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THE 



PRINCIPLES 



OF 



LOGIC, 



HIGH SCHOOLS AND COLLEGES. 



BY 




A. SCHUYLER, LL. D., 

\l 
Professor of Mathematics and Philosophy in Kansas Wesley an University: 
Author of Empirical and Rational Psychology, and of a 
Series of Mathematical Works. 



NEW YORK •:• CINCINNATI •:• CHICAGO 
AMERICAN BOUK COMPANY 






\-, 



so"* 



Copyright, 5869, by Wilson, Hinkle & Co. 
Copyright, 1897, by A. Schuyler. 



SCH. LOGIC. 



^Aan^v ^ 



•I 
i 




PREFACE 



The author has endeavored, in the following treatise to 
give a clear, concise, and systematic development of the 
principles of Logic. 

Care has been taken to retain the valuable results of the 
labors of former investigators — results which the world 
can not afford to lose ; yet much will be found that is new, 
not only in the methods, but also in the matter. 

The works of the following authors have been examined: 
Aristotle, Hamilton, Mill, De Morgan, Thompson, Mansel, 
Whately, Wilson, Tappan, Mahan, Day, McGregor, True, 
and Coppee. To Hamilton, the author is especially in- 
debted for valuable aid in reference to the following sub- 
jects : Classification of Science, General Outline, Concepts, 
and Modified Logic; and to Mill, for examples illustrating 
the four experimental methods of investigation. 

It has been kept steadily in mind that the work is 
designed for a text-book ; and, in accordance with this de- 
sign, a topical arrangement has been given to the matter, 
so as to adapt it to the topical method of conducting reci- 
tations, which, when followed up by appropriate questions, 
is, of all methods, the best for the grade of students who 
will pursue the study of Logic. 

It will add much to the interest and value of the recita- 
tion, to require the student to write out upon the black- 
board, the classifications and all other matter whose con- 
densed form renders it practicable. 



4 LOGIC. 

This practice will induce thorough preparation and 
secure the advantages resulting from the prevalent excel- 
lent methods of conducting mathematical recitations. 

Euler s notation, which has been extensively employed, 
will De found to add greatly to clearness in apprehending 
the principles. This practice has been censured by Mansel 
on the ground that "a concept can not be presented to the 
senses." To this it may be replied that this notation is 
designed to represent, not the concepts themselves, which 
it can not adequately do, but the relation 01 the concepts, 
and this it accurately accomplishes. 

Intuitions have been treated summarily, not in detail, 
and under this head are included not only the ultimate 
facts of reason, but also the ultimate facts of external per- 
ception, called by Kant Sense-intuitions, and the ultimate 
facts of consciousness. 

We invite special attention to the following topics: The 
Fundamental Laws of Thought, Opposition, Conversion, 
Principles warranting (A), (E), (I), (O), Determination 
of the Valid Moods, Discussion of the Figures, Positive 
Propositions and Syllogisms, Induction and Fallacies. 

Some Logicians have discarded the Fourth Figure, on 
the ground that there can be but three, to wit: The mid- 
dle term may be the subject of one premise and the 
predicate of the other, the predicate of both premises, or 
the subject of both. But in the first case there are two 
varieties : the middle term may be the subject of the 
major premise and predicate of the minor, or it may be 
the predicate of the major and subject of the minor. 

In this connection, it is important to observe that the 
major term is simply the predicate of the conclusion, and 
not necessarily greater than the minor term, and that the 
major premise is the premise containing the major term. 
If the Fourth Figure could not be justified on its own 
grounds, there are historical reasons for retaining it. 



PREFACE, 5 

Though the laws of validity warranting the conclu- 
sions (A), (B), (I), (O) have been exhibited (pp. 48-50), 
irrespective of the doctrine of Figure and Mood, yet no 
scholar can afford to remain in ignorance of these subjects 
which display so high a degree of ingenuity and so many 
points of interest. 

The reduction, direct and indirect, of the other figures 
to the first will be found to be not so laborious and dis- 
gusting a process as has sometimes been represented, but, 
if thoroughly done, will prove a most interesting and 
profitable exercise, since it will render the student familiar 
with the Figures, Moods, and Names of the various syllo- 
gisms, as well as show that all can be brought to the test 
of Aristotle's Dictum. 

In Mathematical Induction, general propositions are 
rigorously demonstrated, the premises of the demonstra- 
tion being a particular case ascertained by trial, and the 
demonstrated principle, that if any case is true, the next 
is true. Then, by going back to the case ascertained by 
trial, we have the warrant for concluding that since that 
case is true, the next is true, and the next, and so on ; that 
ts, that the general proposition including all these partic 
ular cases is true. 

Mathematical science, then, has not for its exclusive 
basis, a few fundamental axioms, postulates, and defini- 
tions, as it has so often been represented, but, in addition, 
it includes in its foundation, facts ascertained by experi- 
ence, and from the particular cases, general propositions 
are established. It thus employs Induction as well as 
Deduction, and is relieved from the charge of being an 
exclusively deductive science. 

In this work, the coordinate and subordinate divisions 
and the degree of subordination are indicated by the style 
of numbering. 

The principal divisions arc marked thus, I, II, III, . . . f # 



6 LOGIC. 

the first subdivisions, thus, 1, 2, 3, . . . ; ths second, thus, 
1, 2, 3, . . .; the third, thus, 1st, 2d, 3d, . . . ; the fourth, 
thus, a, b, c, . . .; the fifth, thus, a, /?, /, . . . ; the sixth, 
thus, a', 0', r ' f . , . 

Let the teacher see to it that the student not only un- 
derstands, but that he accurately remembers what he at- 
tempts to learn. The rigid exclusion of all extraneous 
and unnecessary matter, and the systematic arrangement 
and classification of the principles, will enable the student 
readily to commit and retain the whole in his memory. 
His reason will be stimulated by this method; for he will 
thus have in mind, ready for use, the materials for reflec- 
tion, the test of truth, and the safeguard against fallacy. 

A careful preparation on the part of the teacher, thor- 
oughness of drill and repeated reviews, are indispensable 
conditions of success in teaching. 

As the result of both research and thought, the author 
submits the work to the enlightened judgment of his fel- 
low-laborers in the great work of education, hoping that 
it will prove to them a useful auxiliary in imparting a 
knowledge of this most important science. 

A. SCHUYLER. 

Baldwin University, Berea, 0., August, 1869. 




INTRODUCTION. 



I. INTUITIONS. 
1. Definition. 



An Intuition is the immediate perception or appre- 
hension of its object. Intuitions are the elements 
of thought. 

The objects of intuitions are the things immedi- 
ately perceived or apprehended. 

2. Classification. 

1. Empirical or real intuitions; those whose ob- 
jects are perceived as contingent, as the attributes 
of material or spiritual existences. 

1st. Objective; those which are acquired through 
the senses, and which pertain to external phenomena. 

2d. Subjective; those which are acquired by con- 
sciousness, and which pertain to mental phenomena. 

2. Rational or formal intuitions; those whose ob- 
jects are apprehended by reason as necessary. 

1st. Logical; those pertaining to the necessary 
forms of thought. 

2d. Mathematical; those pertaining to the neces- 
sary relations of quantity. 



LOGIC. 



3. Conditions. 



1. Of objective empirical intuitions. 

1st Objective conditions; external phenomena. 
2d. Subjective conditions; the senses — sight, hear- 
ing, touch, taste, smell. 

2. Of subjective empirical intuitions. 

1st Objective conditions; mental phenomena. 
2d. Subjective condition; consciousness. 

3. Of rational intuitions. 

1st. Objective condition ; necessary reality, 

a. Space. 



a. Absolute, 

\0. Time. 



a. Substance. 
b. Conditional, { p. Cause. 



k y. Self-evident relations 
2d. Subjective condition; reason. 



4. Relations of Empirical and Rational Intuitions. 

1. Empirical intuitions are the chronological an- 
tecedents of rational intuitions ; that is, in the order 
of time, empirical intuitions are the first developed in 
the intelligence. 

2. Rational intuitions are the logical antecedents 
of empirical intuitions ; that is, in the order of nature, 
the objects of rational intuitions are the necessary 
conditions of the objects of empirical intuitions. 

5. Order of Evolution. A 

1. Intuitions of concrete objects. 

2. Intuitions of relations. 



INTUITIONS. 



3. Intuitions of relations generalized. 

4. Intuitions as related to intellectual processes. 

5. Intuitions of the infinite and absolute. 



6. Propositions. 

1. Intuitions are realities. 

Proof. 
We are conscious of their existence. 

2. Intuitions are valid. 

Proofs. 

1st. The common sense of mankind asserts their 
validity. 

2d. They harmonize. 

3d. They are free from sources of error. 

Ath. If not valid, our faculties are deceptive, and 
knowledge is impossible. 

bth. Demonstration implies either an infinite series 
of dependent propositions or an ultimate basis. But 
there can be no demonstration by means of an infinite 
series of dependent propositions ; for, to prove one 
proposition by another, and that by another, and so 
on, ad infinitum^ would require infinite time, and is, 
therefore, impossible. Hence, demonstration implies 
an ultimate basis. Now, this basis being ultimate, is 
not derived through any thing else. It must, there- 
fore, be an assumption or an intuition. It can not 
be an assumption; for then it would not be known 
to be true, and might be false, and the demonstration 
would be impossible. The ultimate basis must, there- 
fore, be an intuition, and the validity of demonstra- 
tion implies the validity of intuitions. 



10 



LOGIC. 



7. Corollaries. 

1. The objects of intuitions are realities. 

2. The subject of intuitions is a reality. 

3. The logical antecedents and consequents of in- 



tuitions are valid. 



II. THOUGHTS. 



1. Definition. 

Thought is the recognition of one thing under or 
in another, as a species under its genus, or a property 
in its object. 

2. Processes. 



1. Conceiving. 



2. Judging. 



3. Reasoning. 



1. Concepts. 



" Conceiving is representing a class 
of objects as contained under the 
same attribute, or a combination 
of attributes as contained in the 

^same object. 

c Judging is recognizing the con- 
< gruence or confliction of two ob- 

( jects of thought. 

( Reasoning is deriving one judg- 

\ ment from other judgments. 

3. Products. 

r A concept is the representation of 
a class of objects as contained un- 
der the same attribute, or of a 
combination of attributes as con- 
stained in the same object. 



SCIENCE. 11 

c A judgment is the recognition of 

2. Judgments. < the congruence or confliction of 

(two objects of thoughts. 

r An argument is the derivation of 

3. Arguments. < one judgment from other judg- 

ements. 

4. Contents. 

1. The matter; the objects thought of. 

2. The form ; the manner of thinking. 

1st. The contingent forms ; the phases that may or 
may not appear. 

2d. The necessary forms; the elements that must 
appear, which are subjective, original, universal, and, 
therefore, laws. 

5. Expression. 

1. A term is the expression of a concept in lan- 
guage. 

2. A proposition is the expression of a judgment 
in language. 

3. A syllogism is the expression of an argument 
in language. 

III. SCIENCE. 

1. Definition. 

Science is knowledge classified with respect to prin- 
ciples. 

2. Classification. 

1. Direct science; science of objects. 



12 LOGIC. 

1st. Science of external phenomena; physical sci- 
ence. 

2d. Science of internal phenomena; mental science. 

2. Reflex science; science of sciences — Logic. 

1st. Subjective, formal, abstract, or pure logic, whicl 
treats of the laws under which the human mind can 
know; the conditions of knowledge which lie in the 
nature of thought itself; the relation of thought to 
its object — the logic of Aristotle. 

2d. Objective, material, concrete, or applied logic, 
which treats of the laws under which an object can 
be known ; the conditions of knowledge which lie in 
the nature of the objects of thought; the relation ol 
the object to the thought — the logic of Bacon. 




LOGIC. 



I. GENERAL OUTLINE. 

1. Definition. 

Logic is the science which treats of the formal laws 
of human thought, 

2. Exposition. 

1. Its etymology. The word Logic is derived from 

Xoyixyj an adjective, intery/iT] Science, rtyvr) art, Or Tipay^areia 

investigation, being understood. But Xoyixij is from 
Xdyoq^ a word ambiguous in its import, denoting both 
thought and the expression of thought, and thus 
equivalent both to the Latin ratio and oratio. 

2. Its genus. Logic is a science rather than an art. 

3. Its province. The province of Pure Logic is the 
formal laws of human thought, both general and 
special; general, when it treats of the fundamental 
laws of thought; special, when it treats of the laws 
applicable to concepts, judgments, or arguments. 

3. Classification. 

1. In reference to the mind, Logic is classified as 
1st. Systematic; the complement of doctrines con- 
stituting the science. Systematic logic is objective. 

13 



14 LOGIC. 

2d. Habitual ; a knowledge of the science and skill 
in its application. Habitual logic is subjective. 

2. In reference to its application, Logic is classi- 
fied as 

1st. General or Abstract, which treats of the formal 
laws of thought without reference to any particular 
matter, and embraces 

a. Pure Logic, which treats of the formal laws of 
thought as contained, a priori, in the nature of the 
intelligence itself, embracing 

a. The doctrine of elements, relating to 

a. The fundamental laws of thought or the univer- 
sal conditions of the thinkable. 

/9\ Special laws, relating to concepts, judgments, or 
arguments. 

/5. The doctrine of method, relating to 
a. The general laws of method. 
/?'. Special laws, relating to definition, division, anal- 
ysis, or proof. 

6. Modified Logic, which treats of 
a. The nature of truth and error, and the laws of 
their discrimination. 

/?. The causes of error and the impediments to the 
attainment of truth, which are 

a. Physical ; disease, hunger, thirst, peculiarities of 
temperament. 

p. Physico-mental ; imperfection in the senses. 

y. Mental ; weakness or derangement of the menta' 
faculties — the intellect, the sensibilities, or the will. 

&. Circumstantial ; education, rank, age, national 
lty, social relations, etc. 

y. Aids to correct thinking, embracing 



CONGRUENCE, CONFLICTION, OPPOSITION. 15 

a. The acquisition of knowledge, in the various 
ways. 

p. The communication of knowledge. 

2d. Special or Applied, the methodology of the 
various sciences. 



II. CONGRUENCE, CONFLICTION, OPPOSITION. 
1. Definitions. 

1. Concepts that can be united in thought are con- 
gruent. 

2. Concepts that can not be united in thought are 
conilictive or opposed. Opposition is of two kinds : 

1st. Contrary opposition, m which two objects not 
universally inclusive, are mutually repugnant. Thus, 
red and blue, walking and standing, etc. 

2d. Contradictory opposition, in which two objects 
together are, within their sphere, universally inclusive 
and mutually repugnant. Thus, red and not-red, 
walking and not- walking, honest and dishonest within 
the sphere of moral beings. 

2. Classification. 

Concepts. {J; CoXtivel'. / If Contraries. 

I 2d. Contradictories. 

3. Formulas. 

1. For contraries : P is not Q. 

2. For contradictories : Non-P is Q. 



16 



LOGIC. 



HI. FUNDAMENTAL LAWS OF THOUGHT. 
1. The Laws of Identity. 



1. Laws, 



1st A concept and the sum of all of its 
elements are totally identical. 
2d. A concept and a part of its elements 
^are partially identical. 



™ , (1st. C=e+e f +e" ..., , 
2. Formulas.) > \ if 

I 2d. Some (7=6, 



C denotes the 

concept, 

and g, e\ e h \ . . . , 

its elements. 



r lst. C may be substituted for e+e'+e" 
o ^ 77 . J + . . . , and conversely. 

| 2<i. Some C may be substituted for e, 
^and conversely. 

2. The Law of Conflicthes. 

( Conflictives can not be affirmed of the 
\ same object. 

8 is not both P and Q, if P is not Q, 
or if non-P is Q. 

c 1st. If SisP,S is not §, ) f P is not Q, 
3. Corollaries. < v if ^ or 

1 2d. If ShQ,S is not P, J I non-P is Q. 

Scholium. — The Law of Conflictives is usually called the Law of 
Contradiction. 



1. Law. 

2. Formula. 



1. Law. 



3. The Law of Contradictories. 

One of two contradictories must be 
affirmed and the other denied. 






CONCEPTS. 17 



2. Formula. S is either P or Q, if non-P is Q. 
1st. If /Sf is P, /Sis notQ," 



8. Corollaries. { ' T „ ~. , -« c,. ^ ^ if non-P is 0. 
3a. If # is not P, # is Q, ^ 

,4th. If ^is not $, £ is P„ 

Schclium. — The Law of Contradictories is usually called the Law of 
Excluded Middle. 

4. The Law of Reason and Consequent. 

1. Law. — A reason implies a consequent. 

2. Formula. — P implies C, if P is the reason of C 



3. Corollaries. 



1st. If Pis, (7 is. 

2d. If C is, some P is. 

3<i. If every P is not, C is not. 

4tth. If (7 is not, P is not. 

IV. CONCEPTS. 

1. Definition. 

A concept is the representation of a class of objects 
as contained under the same attribute, or of a combi- 
nation of attributes as contained in the same object. 
A concept is the product of which conception is 
the act. 

2. Etymology. 

Concept, conception, from concipio [con y capio\ signi- 
fies comprehending many into one. 

3. Nature of the Elements. 

1. The immediate and irrespective knowledge of an 
object by intuition. 

l. 2 



18 LOGIC. 

2. The mediate and relative knowledge of the 
object as comprising attributes common to it with 
other objects. 

4. Formation. 

1. A plurality of objects is furnished by intuition. 

2. These objects are compared, their resemblances 
and differences noted. 

S. By attention, the thoughts are directed to the 
similar and abstracted from the dissimilar. 

4. The similar objects are combined into an exclu- 
sive object of thought. 

5. Relation to Language. 

Concepts, in order to become available, must be em- 
bodied in a verbal sign. 

Language is the product, the instrument, and the 
embodiment of thought. 

6. Characteristics. 

1. Inadequacy. — A concept is inadequate or incom- 
plete, since but a part of its elements can be repre- 
sented in thought. Thus, the concept animal does not 
actually represent to the mind all the subordinate 
classes contained under it. The concept man does not 
represent all the attributes of an individual, since it 
must represent only the attributes common to all 
human beings. 

2. Relativity. — Concepts, as the result of comparison, 
are necessarily relative. They afford no absolute or 
irrespective object of knowledge, and can only be re- 
alized in consciousness by applying them as terms }f 



CONCEPTS. 19 

relation to one or more of the objects which agree in 
certain points of resemblance which they express. 

3. Potential universality. — A concept can not be 
represented as a universal in the imagination, for this 
would require the representation of conflictive attri- 
butes as applied to the same object. 

A concept is universal, not because it represents, at 
once, all the objects of a class, but because it may so 
vary as to represent any. 

7. Classification as to Quantity. 

1. Extensive concepts are representations of classes 
of objects as contained under the same attribute. 

2. Comprehensive concepts are representations of 
combinations of attributes as contained in the same 
object. 

8. Extension and Comprehension. 

1. The extension of a concept varies inversely as its 
comprehension ; that is, the greater the extension, the 
less the comprehension, and conversely. 

Thus, the concept animal is greater, as to extension, 
than the concept horse, since it contains under it the 
concept horse, as a species, together with a great variety 
of other species. It is less as to comprehension, since 
it contains in it, as attributes, those attributes only 
which are common to all the species contained under 
it, whereas the concept horse contains all the attributes 
common to all these species, together with what is 
characteristic of itself. 

2. A simple concept, that is, a concept not involving 
a plurality of attributes, is a maximum as to extension, 
and a minimum as to comprehension. 



20 LOGIC. 

Thus the simple concept being, which is the highest 
genus, is a maximum as to extension, since it contains 
under it all other classes. It is a minimum as to com- 
prehension, since it contains no attribute which is not an 
attribute of every class and individual contained under 
it. Hence, it is extensive rather than comprehensive. 

3. A simple concept is capable of division, since it 
can be resolved into classes and individuals. 

It is incapable of definition-; for an object is defined 
by referring it to the class immediately containing it, 
and distinguishing it from other objects of the class 
by means of its characteristic or differential quality.; 
But the simple concept, or highest genus, is not con 
tained under a higher genus, nor has it a differential 
quality. It can not, therefore, be defined. 

It is incapable of analysis, since it does not contain 
a plurality of attributes. 

4. An individual concept, that is, a concept not; 
involving a plurality of objects, is a minimum as to 
extension, and a maximum as to comprehension. 

Thus, an individual concept, containing no classes 
or individuals under it, is a minimum as to extension. 
It is a maximum as to comprehension, since it contains 
all the attributes common to all the individuals of its 
class, together with what is characteristic of itself. 
Hence, it is comprehensive rather than extensive. 

5. An individual concept is capable of definition, 
for it is contained under a class and has a differential 
quality. It is incapable of division, since it contains 
nothing under it. It is capable of analysis, since it 
contains a plurality of attributes. 

6. A concept, neither simple nor individual, is 
neither a maximum nor a minimum, either as to ex- 
tension or comprehension 



CONCEPTS. 21 

Since it is neither the highest genus nor an individ- 
ual, it is neither a maximum nor a minimum as to ex- 
tension. Since the higher the genus, the less the 
comprehension, and the lower the species, the greater 
the comprehension, it is neither a maximum nor a 
minimum as to comprehension. Hence, it is both 
extensive and comprehensive. 

7. A concept neither simple nor individual is capable 
of definition, since it is contained under a class, and 
has a differential quality. It is capable of division, 
since it contains either classes or individuals under it. 
It is capable of analysis, since it contains a plurality 
of attributes. 

9. Classification as to Quality. 

1. Clear concepts are those which are discriminated, 
as a whole, from other concepts. Clearness is attained 
by definition. 

2. Obscure concepts are those which are confounded 
with other concepts. 

Obscureness is avoided by definition. 

3. Distinct concepts are those in which the classes 
or individuals contained under them, or the attributes 
contained in them, are discriminated. 

Extensive distinctness is attained by division, com- 
prehensive distinctness is attained by analysis. 

4. Indistinct concepts are those in which the classes 
or individuals contained under them, or the attributes 
contained in them, are confounded. 

10. Qualities involved in Distinctness. 

1. A clear apprehension of the classes or individuals 
contained under the concept, or of the attributes con- 
tained in it. 



22 LOGIC. 

2. A clear discrimination of these classes or indi- 
viduals and attributes. 

3. A clear recognition of the nexus which binds 
these classes or individuals and attributes into unity. 



11. Concepts admitting Extensive or Comprehensive 
Distinctness. 

1. Simple concepts are capable of extensive, but in 
capable of comprehensive, distinctness. 

2. Individual concepts are capable of comprehen- 
sive, but incapable of extensive, distinctness. 

3. Concepts, neither simple nor individual, are capa- 
ble both of extensive and comprehensive distinctness. 

12. Specific Rules for attaining Distinctness. 

1. Seek for the positive elements; the negative may 
be sought for as aids in determining the positive. 

2. Among the positive elements, seek out the intrin- 
sic and permanent in preference to the extrinsic and 
transitory. 

3. Among the intrinsic and permanent, seek out the 
necessary and essential, then descend to the contingent 
and accidental. 

13. Sources of Indistinctness. 

1. The nature of the concept itself, which is multi- 
plicity, bound, by a mental process, into unity. 

2. The dependence of the concept on language as 
the condition of its continuance. 

14. Remedy for Indistinctness. 

The remedy for indistinctness is the rules for dis- 
tinctness. 



CONCEPTS. 23 

15. Classification as to Validity. 

1. Valid concepts are those which embrace only in- 
tuitions, or intuitions with their logical antecedents 
and consequents. Valid concepts are true. 

2. Invalid concepts are those which embrace as- 
sumed elements. Invalid concepts may be true or false. 

16. Classification as to Truth. 

1. True concepts are those which correspond to their 
objects. True concepts may be valid or invalid. 

2. False concepts are those which do not correspond 
to their objects. False concepts are invalid. 

17. Classification as to Congruity. 

1. Congruous concepts are those in which all of 
the elements harmonize. 

Congruity depends on the nexus which binds the 
elements together into unity. 

2. Incongruous concepts are those which embrace 
sonflictive elements, — contraries or contradictories. 

Incongruity is the mark of invalidity and indicates 
chat some elements have been assumed without war- 
rant. 

18. Classification as to Completeness. 

1. Complete concepts are those which embrace all 
of the elements of their objects. 

Completeness is, in general, an ideal perfection. 
Very few, if any, of our concepts are complete. 

2. Incomplete concepts are those which embrace 
only a part of the elements of their objects. 

Incompleteness characterizes most, if not all, of our 
concepts. Most objects have qualities which have 



24 LOGIC. 

escaped our observation. Different persons form dif- 
ferent concepts of the same objects, one combining one 
set of elements, another, another set. 

19. Classification as to their Relations in Extension. 

1. Of Inclusion. — One concept is included in another 
when the sphere of the former is contained in the 
sphere of the latter. 

There are two cases of inclusion : 

1st. Of Subordination. — One concept is subordinate 
to another w^hen the former is contained under the 
latter as a species under a genus, or as an individual 
under a species. Thus, the concept horse is subordi- 
nate to the concept quadruped, since horse is a species 
of which quadruped is the genus. The concept George 
Washington is subordinate to the concept man, since 
George Washington is an individual of which man is the 
species. 

If one concept is subordinate to another, it is subor- 
dinate to any higher concept. Thus, since the species 
horse is subordinate to the genus quadruped, it is sub- 
ordinate to any higher genus, as animal, organized 
being, being. 

2d. Of Coextension. — One concept is coextensive with 
another when they have a common sphere. 

Thus, equilateral triangles are coextensive with equi- 
angular triangles, since they have a common sphere, 
that is, since every equilateral triangle is equiangular, 
and every equiangular triangle is equilateral. 

2. Of Exclusion. — One concept is excluded from an- 
other when their spheres have no part common. 

There are two cases of exclusion : 

1st. Of Coordination. — Two concepts are coordinate 






CONCEPTS. 25 



when they are exclusive and both immediately com- 
prehended under the same concept. Thus, the con- 
cepts horse and sheep are coordinate, since they are ex- 
clusive, and both immediately subordinate, as species, 
to the genus quadruped. 

2d. Of Non-coordination. — Two exclusive concepts 
are non-coordinate when they are not immediately 
subordinate to the same concept. Thus, the concepts 
Arabian horse and sheep are non-coordinate, since 
Arabian horse is a species of the genus horse, while 
sheep is not. 

3. Of Intersection. — Two concepts intersect when 
their spheres have a common part, and each, a part 
not common. Thus, the concepts men and liars inter- 
sect, since some men are liars and some liars are men, 
some men are not liars and some liars are not men. 

20. Notation expressing these Relations. 

1. The relation of subordination may be expressed 
by one circle within another, the larger circle express- 
ing the superior concept, the smaller, an inferior. 

Thus, (5) 

2. The relation of coextension may be expressed 
by two equal coincident circles. 

Thus, 

3. The relation of coordination may be expressed by 
two equal exclusive circles both contained within a 
larger circle. The larger circle expresses th«* superior 

L. 3 







26 LOGIC. 

concept, the smaller circles express coordinate inferior 
concepts. 

Thus, HO 

4. The relation of exclusive non-coordination may 
be expressed by one of two circles within, and the 
other without, a third circle. 

Thus, (g)0 

5. The relation of intersection may be expressed by 
two intersecting circles. 




Thus, 

21. Summary of the Relations of Extensive Concepts. 

fist. Subordination. 



1. Inclusion. 

2d. Coextension. 



© 




2. Exclusion. 



r lst Coordination. |o Q) 
J2d. Non-coordination, f Cj ) f~^\ 



3. Intersection. 




CONCEPTS. 



27 



22. The Laws of Classification by Genera and Species. 

1. The Law of Homogeneity. — However different two 
concepts, neither of which is the highest genus, both 
are subordinate to the same higher concept, though 
not necessarily coordinate with each other; for, ulti- 
mately, every concept may be referred to that of being, 
the highest genus. Hence, things the most dissimilar, 
must, in certain respects, be similar. 

2. The Law of Heterogeneity. — Every concept con- 
tains other concepts under it. In thought, therefore, 
the division of concepts gives concepts, not individ- 
uals. Hence, things the most similar must, in certain 
respects, be dissimilar. Thus, take any two concepts 
with a small difference. Now, this difference can be 
divided, thus giving new concepts distinguished by 
this partial difference, and so on, ad infinitum. But 
the infinite divisibility of concepts, like the infinite 
divisibility of space, time, and matter, exists only in 
speculation. 

To illustrate, let us classify angles thus : 



f Ki gK 



Angles, < 



Acute, 



Oblique, < 



Obtuse, 



Here we pause, not because it is impossible to pursue 
the classification farther, but because it is not called 



28 



LOGIC. 






for. But we can conceive these angles situated in a 
horizontal, a vertical, or an oblique plane, giving hori- 
zontal, vertical, or oblique angles, and these angles 
may have any position in these planes, and the sides 
may take an infinite number of directions, for each 
position of the vertex, and the acute angle may vary 
through an infinite number of states between the lim- 
its and 90°, and the obtuse angle may vary, in like 
manner, between the limits 90° and 180°. 

23. Classification as to Relations in Comprehension. 

1. As to Identity. 
c a. Absolutely identical. 



1st. Identical. 

2d. Different. 

1st. Congruent. 
2d. Conflictive. 



b. Relatively c a. Reciprocating 
identical. 1 or convertible. 

I /?. Similar or cognate 



( a. Absolutely different. 
\ b. Relatively different. 

2. As to Congruence. 



a. Contrary. 

b. Contradictory. 

3. As to Elements. 

1st. Intrinsic ; those formed of essential elements. 
2d. Extrinsic; those formed of accidental elements. 

4. As to Proximate Relations. 

1st. Of Involution. — One concept is involved in an* 
other, when the first forms a part of the sum-total of 
the elements which together constitute the conipre- 



JUDGMENTS. 29 

nension of the second. Thus, the sides, angles, and 
area are involved in the concept, triangle. 

2d. Of Coordination. — Two concepts are coordinate 
when they are exclusive, and both immediately com- 
prehended, as elements, of the same concept. Thus, 
the sides and angles of a triangle are coordinate. 

V. JUDGMENTS. 
1. Definition. 

A judgment is the recognition of the congruence 
or confliction of two objects of thought, 

2. Expression. 

A proposition is the expression of a judgment in 
language. Thus, S is P. 

3. Elements. 

1st. The subject, or determined 

! mi , concept — S. 

1. 1 he concepts. { „ _ _, r _. ■ _ t 
2d. The predicate, or determining 

concept — P. 

2. The relation of the concepts — in the copula, is. 

4. Concepts and Judgments compared. 

1. A concept may be regarded as an implicit or un- 
ieveloped judgment. 

2. A judgment may be regarded as an explicit or 
developed concept. 



30 L.OGIC. 



5. Classification as to Origin. 



1. Primitive. < 



' 1st. Assumptive. 

' a. The predicate the log- 
ical antecedent of the 
subject. 
b. The predicate an es- 
sential attribute of the 
subject. 






2d. Intuitive. < 



tive. { 

I: 



i 1st. Problematical. 
2. Derivative. { _, 

2a. Demonstrative. 



6. Classification as to Validity. 

{ls£. When the concepts are valid, and 
2d. The relation r a. Intuitive, or 
of the concepts is \ b. Demonstrative. 

2. Invalid; when the laws of validity are violated. 

7. Classification as to Truth. 

1. True ; when the relation expressed corresponds to 
the reality. 

2. False ; when the relation expressed does not cor- 
respond to the reality. 

8. Classification as to Extension and Comprehension, 

1. A judgment is extensive when the determining 
predicate is considered as the whole of extension con- 
taining the subject. Thus, man is an animal. 



JUDGMENTS. 31 

2. A judgment is comprehensive when the deter- 
mined subject is considered as the whole of comprehen- 
sion containing the predicate. Thus, man is mortal. 

9. Classification as to Form. 

1. A categorical judgment is one in which the rela- 
tion of the subject and predicate is unqualified by a 
condition. Thus, S is P. 

2. A conditional judgment is one in which the rela- 
tion of the subject and predicate is qualified by a 
condition. 

Conditional judgments are of three varieties : 

1st. Hypothetical, when the qualifying condition is 
an hypothesis. Thus, if S is P, T is U. 

2d. Disjunctive, when the qualifying condition is 
a disjunction. Thus, S is either P or Q. 

8d. Dilemmatic, when the qualifying condition is 
both an hypothesis and a disjunction. Thus, if S is P, 
T is either U or V. 

10. Classification as to Quantity. 

1. Universal, iff"' 

JN o S is P. 



{ 



Some S is P. 
2. Particular. { 

borne o is not P. 



11. Classification as to Quality. 

1. Affirmative. < 



2. Negative. \ 



Some S is P. 

No S is P. 
Some S is not P. 



32 LOGIC. 

12. Principles of Expression. 

1. Principles warranting Affirmation. 

1st. Immediate. 

a. Principles. — The Laws of Identity. 

6. Formula. — S is P, if S and P are identical, either 
a. Totally, or p. Partially. 

2d. Mediate, 
a. Principle. — The Law of Contradictories. 

, fa. £isP, if Sis not Q,\ .„ „. ^ 

6. Formulas. 1 « a • r* -.p cv + r> r " non-P is 0. 
( /?. o is ^, it aS is not P, J ^ 

2. Principles warranting Negation. 
1st. Immediate. 

a. Principle. — The Law of Conflictives. 

b. Formula. — P is not Q, if P an( i Q are conflictives. 

2d. Mediate. 

a. Principle. — The Law of Conflictives. 

, ™ 1 r a. S is not P, if £ is Q,] if Pis not $, or 

b. * ormulas. j ^ . g ^ ^ . f ^ ig p? | 7 ^_ p ig Q 

3. Principle warranting Hypothecation. 

1st. Principle — The Law of Reason and Consequent. 

2d. Formula — If P is, C is, if P is the reason of C. 

4. Principle warranting Disjunction. 

1st. Principle. — The Law of Contradictories. 

7 J* 1 ( S is either P or 0, if P and Q are con- 

2a. r ormula. < ^' x 

I tradictones. 



JUDGMENTS. 



33 



13. Classification of Categorical Judgments. 

1. Universal. { Jf Affirmative. (A) All 8 is P 
X 2d. Negative. (E) No S is P. 



2 P t" 1 I ^' Affirmative. (I) Some $ is P 
' I 2c?. Negative. (0) Some S is nc 



(0) 



not P. 



14. Laws of Validity. 

1. (A) All £ is P. 



Valid in case of 
Inclusion. 



1st. Subordination. 



2d. Coextension. 





2. (E) No 8 is P. 
Valid in case of Exclusion. 

3. (I) Some S is P. 
1st. Intersection. 



Valid in case of 




2d. Inclusion. 



a. Subor- 
dination. ' 



b. Coex- 
tension. (s P] 





r(©' 



S P 



34 



LOGIC. 

4. (0) Some S is not P. 
' 1st. Intersection. 



Valid in case of< 



2.- 



3.- 



4.- 



2d. Inclusion — Subor 
dination. 

Zd. Exclusion. 



15. Opposition. 

(A ) Contraries. ( E) 





1st. 

.2d. 

1st. 

.2d. 
'1st. 

.2d. 

■1st 

.2d. 



(I) Sub-contraries. (O) 

f a. The truth of (I) 
The truth of (A) implies j k The falgity of j (E) 

The falsity of (A) implies the truth of (0) 

f a. The truth of (O) 
The truth of (E) implies j ^ The falgity of j (A) 

The falsity of (E) implies the truth of (I) 

The truth of (I) implies the falsity of (E) 

a. The truth of I ( E ) 



The falsity of (I) implies J " " { (0) 

( b. The falsity of (A) 
The truth of (O) implies the falsity of (A) 

™ -,. *,~. ,. fa- The truth of { ^ 
The falsity of (0) implies J I (I) 

\b. The falsity of (E) 



JUDGMENTS. 35 

16. The Laws of Opposition. 

1. The truth of a universal implies the truth of its 
particular. 

2. The falsity of a universal does not imply the 
falsity of its particular. 

3. The falsity of a particular implies the falsity of 
its universal. 

4. The truth of a particular does not imply the truth 
of its universal. 

5. The contraries can not be both true, but may be 
both false. 

6. The sub-contraries can not be both false, but may 
be both true. 

7. Two contradictories can not be both true or both 
false. 

17. Distribution of the Concepts of a Judgment. 

1. Definitions. 

1st. A concept is distributed when all of it is taken. 
2d. A concept is undistributed when only a part of 
it is taken. 

2. Principles. 

1st. All universals distribute the subject. 
2d. All negatives distribute the predicate. 
3d. No particular distributes the subject. 
ith. An affirmative may or may not distribute the 
predicate. 

3. Consequences. 

1st. (A) distributes the subject, and may distribute 
the predicate. 

2d. (E) distributes both the subject and predicate. 



36 LOGIC. 

3d. (I) does not distribute the subject, but may din- 
tribute the predicate. 

4th. (0) distributes the predicate, but not the subject. 

4. Remarks. 

1st. (A) distributes the predicate in case S and P are co- 
extensive. In this case, all S is all P, and all P is all S. 

This relation holds true in case of definitions, equa- 
tions, identical propositions, and when S and P desig- 
nate, by different attributes, the same class not suscep- 
tible of subdivision. Thus, All equilateral triangles are 
equiangular, and All equiangular triangles are equi- 
lateral. 

2d. (I) distributes the predicate in case S is the 
genus of which P is a species, or in case S is the 
species of which P is an individual. Thus, Some ani- 
mals are all horses. 

3d. Both (E) and (0) distribute the predicate. Thus, 
No S is P means Any S is not any P. Some S is not P 
means Some S is not any P. 

4th. For those cases in which the predicate of a 
negative is, by an express statement, undistributed, 
see Hamilton's Classification of Propositions. 

18. Conversion. 

1. Definition. 

The converse of a proposition is the proposition 
obtained by transposing the terms of the original 
proposition. Thus, the converse of S is P is P is S. 

2. Kinds. 

1st. Simple conversion, when the converse has the 
same quantity and quality as the original proposition. 
a. (E) may always be converted simply. 



JUDGMENTS. 37 

Thus, (E) No 8 is pY s V p J(E) No P is S. 

b. (I) may always be converted simply. Thus, 
B) Some S is P. ( s A p J H0p) (si M (s *) 

Conversely, (I) Some P is S. 

<?. (A) may be converted simply when S and P are 
coextensive. 

Thus, (A) All S is P. U pW) All P is S. 

d. (A) may always be changed to (E) which may be 
converted simply. Thus, 



(A) All S is P = (E) No S is non-P. 
Conversely, (E) No non-P is S. 



0p) (CJ) 



e. (0) may be changed to (I) which may be con- 
verted simply. 
Thus, (0) Some S is not P = (I) Some S is non-P. 



®9^®)00 (i) 



Some non-P is S 

2d. Conversion by limitation^ when the quantity is 
reduced in the converse. 

a. (A) may ahvays be converted by limitation. 
Thus, (A) All S is P. U A f p) (I) Some P is S. 



38 



LOGIC. 



b. (E) may always be converted by limitation. 

(0) Some P is 



Thus, (E) No S is P. s 





not S. 



e. (E) may always be changed into A, which may be 
converted by limitation. 



Thus, (E) No Sis P^ 
(A) All S is non-P. 





(I) Some non-P 

isS. 



19. Classification of Hypothetical Judgments. 

Forms. Laws of Validity. 



1. If A is B, A is C. 



If B is C. 




2. If A is B, A is not C. If no B is c/Qb) f c 

3. If A is not B, A is C. { If B . and C are «>^adic 

I tones. 

1. IfA is not B, A is notC. IfCisB. f a j (0 B y 
5. If A is B, Bis C. 



!If A is C and co- 
extensive with f (abIc] 
B. 
r If A is not C 



6. If A is B, Bis not C 

7. IfA is not B, Bis C. f 



and coex 







tensive y J 
with B. ^-^ 
If A and C are contradic- 
tories. 



JUDGMENTS. 



39 



8. IfA is not B, Bis notC. If C is A. ( (oJa) ( b 

(If C is B, and A 



9. If A is B, Cis A. 



coextensive (amT)b) 
with B. 



10. If AisB, Cis not A. If CisnotB. f QbJ ( c 

11. If A is not B, Cis A. | 



If A and B are contradic- 
tories, and C is not B. 



12. If A is not B, C is not A. If C is B. 



13. If AisB, Cis B. IfCisA. 



© o 




If A is coex- 
tensive with 






-. a T* a • t» ^ • , -r> tensive with / \ 
14. If A is B, Cis not B.^ and Q .U b) 



not A. 







15. If A is not B, C is B. 



If A and B are contradicto- 
ries, and C is not A. 



16.IfAisnotB,CisnotB. If C is A. 



17. If A is B, C is D. / Ij L C is J 

I BisD 



A and 




40 LOGIC. 



<ioti?a -on- ,-p.fIfCisA and ^-^ 
18. If A is B, C is not D. { _ f^\ 

I B is not V. \U 







-i a Ti? a 4. d r« • -n f If A is B and C is D are con- 

19. If A is not B, C is V. J 

( tradictory propositions. 

20. If A is not B, C is (if C is A and /^\ /^\ 

notD. I DisB. (^) K^ B ) 

20. Classification of Disjunctive Judgments. 

The judgment, S is P or Q, may be 

1. Divisive. Thus, Angles are right or oblique ; that 
is, angles are divided into right angles and oblique 
angles. 

2. Disjunctive in expression. Thus, This electricity is 
vitreous or positive — these terms denoting the same 
kind of electricity. 

3. Disjunctive in thought. Thus, The animal is a ver- 
tebrate or an invertebrate. 

The first and second classes are disjunctive only in 
form, but categorical in sense. The third class only is 
logically disjunctive. This disjunction may be 

1st. In the copula. Thus, S either is or is not P. 
This is pure contradictory opposition, and the judg- 
ment is valid by the Law of Contradictories. 

2d. In the terms. Thus, S is either P or Q. In this 
case, there are two varieties : 

a. When P and Q are contraries. We then have con- 
trary opposition, and can not affirm a priori, but only 
a posteriori, that S is either P or Q. 



JUDGMENTS. 



41 



In contrary opposition, the judgment, S is either 
P or Q, is valid on the condition that all suppositions 
contrary to it are shown to be false. Then, 

If S is P, S is not Q ; if S is Q, S is not P ; 

If S is not P, S is Q ; if S is not Q, S is P. 

b. When P and Q are contradictories. We then have 
contradictory opposition and the judgment is valid by 
the Law of Contradictories. 

The disjunction may lie in the subject as well as in 
the predicate, when the disjunction is in the terms. 



21. Classification of Logical Disjunctives. 



r 1. In the copula — Pure contr 
dictory. 



o 
o 






2. In the terms. < 



8 either is or is 
not P. 
'a. In the subject. 
Either R or S 

r 1st Contrary. < , T ',, -,. 

J l 6. In the predi- 
cate. S is either 
P or Q. 
a. In the subject. 
Either S or non- 
2d. Contra- g i s p. 

L dictory. ^ b. In the predi- 

cate. S is either 
P or non-P. 



22. Classification of Dilemmatic Judgments. 



1. 



1. If A is B, S is either P or Q. 

2. If A is B, S is neither P nor Q. 

3. If A is not B, S is either P or Q. 

4. If A is not B, S is neither P nor Q. 

L. 4 



42 LOGIC. 

'5. If either A is B or C is D, S is either P or Q. 

6. If either A is B or C is D, S is neither P nor Q 

7. If neither A is B nor C is D, S is either P or Q 

8. If neither A is B nor C is D, S is neither P nor Q 

The disjunctive consequent is in contrary opposition, 
for, if in contradictory opposition, the consequents of 
1, 3, 5, and 7 would be true, and the consequents of 2 
4, 6, and 8 would be false, by the Law of Contradicto 
ries, and the reason in the conditional clause would be 
redundant. Thus, it would be superfluous to say, If A 
is B, S is either P or non-P; for S is either P or non-P, 
whether A is B or not. 

VI. ARGUMENTS. 

Definition. 

An argument is the derivation of a judgment from 
another judgment or from other judgments. 

1. IMMEDIATE ARGUMENTS. 

1. Definition. 

An immediate argument is an argument in which 
the relation of the concepts of the derived judgment 
is inferred from another judgment or from other judg- 
ments without the intervention of a middle concept. 

2. Tarieties. 

1. Inferences from opposition. [V. 15.] 

2. Inferences from conversion. [V. 18.] 

3. Inferences from modal restriction. 

Thus, S is necessarily P, .-. S is actually P, .\ S is 
probably P, .-. S is possibly P. 



ARGUMENTS. 



43 



4. Inferences from composition. 

Thus, A is in C, B is in C, .*. A and B are in C. 

5. Inferences from divisive judgments. 



Thus, S is P, Q, or K, .-. < 



(1st. The P of S is neither 
the Q nor the R of S. 

2d. The non-P of S is either 
the Q or the R of S. 

Sd. The neither P nor Q of 
S is the Pv of S. 



6. Inferences by means of privatives. 
'Isf.AllSisP 



<±H 



2d. All non-S 
is non-P. 



(E) 






r Is*. No S is P. 



2d. No non-S is 
non-P. 

1st. Some S 
is P. 




a. No S is non-P. 

b. All non-P is 
non-S. 

a. No non-S is P. 



(3D 



OH 



2d. Some non-S 
is non-P. 




b. All P is S. 

a. All S is non-P. 

b. All P is non-S. 

a. All non-S is P. 

b. All non-P is S. 

r a. Some S is not 

non-P. 
b. Some P is not 
non-S. 

a. Some non-S is 
not P. 

b. Some non-P is 
not S. 



44 



LOGIC. 



(O) 



1 



1st Some S is ( s 
not P. 




2d. Some non-S 
is not non-P. 




a. Some S is non-P. 

b. Some non-P is 
not non-S. 

a. Some non-S is P. 
6. Some P is not S. 



2. MEDIATE ARGUMENTS. 



1. Definitions. 

1. A mediate argument is an argument in which the 
relation of the concepts of the derived judgment is in- 
ferred from other judgments through the intervention 
of a middle concept. 

2. The derived judgment is called the conclusion. 

3. The judgments from which the derived judgment 
is inferred are called the premises. 

2. Expression. 

A syllogism is the expression of an argument. 

■ Hence, a syllogism contains 



Thus, 



(All Mis P. 

1 All S is M. 
I. -.All Sis P.. 



1. Three 
terms. 



1st. The ex- 
tremes. 



' a. The major term (P), the 
predicate of the conclusion. 
b. The minor term (S), the 
subject of the conclusion. 
2d. The middle term (M), the medium 
of comparison. 



ARGUMENTS. 



45 



2. Three 

proposi- ^ 
tions. 



1st. The 

premises 



a. The major premise in 
which (M) and (P) are com- 
pared. 
^ The minor premise in 
which (S) and (M) are com- 
pared. 

2d. The conclusion in which the relation 
of (S) and (P) is inferred. 

3. Illustration. 



Every responsible agent is a free agent. 

Man is a responsible agent. 

.-. Man is a free agent. 

The subject, man, and the predicate, a free agent, of 
the conclusion, Man is a free agent, are the extremes, 
of which the predicate, a free agent, is the major term, 
and the subject, man, is the minor term. 

The term, responsible agent, with which the ex- 
tremes are separately compared in the premises, is the 
middle term. 

The premise, Every responsible agent is a free 
agent, in which the middle term is compared with the 
major term, is the major premise. 

The premise, Man is a responsible agent, in which 
the minor term is compared with the middle, is the 
minor premise. 

The middle term is found in each of the premises, 
but not in the conclusion. 

One extreme is found in one premise, the other in 
the other, and both in the conclusion. 

Let us now consider this argument, as an argument, 

1. In extensive quantity. In this case, the concept, 
responsible agent, is contained under the concept, 



46 LOGIC. 



' 



free agent; that is, the class of responsible agents 
is a species of which the class of free agents is the 
genus. 

The concept, man, is contained under the concept, 
responsible agent; that is, the class, man, is a species 
of which the class of responsible agents is the genus. 

Hence, on the principle, that a part of a part is a 
part of the whole, the concept, man, is contained un- 
der the concept free agent. 

In extensive quantity, the copula, is y signifies is con- 
tained under. 

Let us now generalize and symbolize this argument, 
thus: 

All M is P. //^y\ All M is contained under P. 
All S is M. ( [®yj All Sis contained under M. 
/.All S is P. v~^/ .-. All S is contained under P. 



2. In comprehensive quantity. In this case, the con- 
cept, responsible agent, contains in it, that is, compre- 
hends, the concept, free agent, as one of its attributes 

The concept, man, comprehends the concept, respon- 
sible agent, as one of its attributes. 

Hence, on the principle that the whole comprehends 
a part of a part, the concept, man, comprehends the 
concept, free agent. 

In comprehensive quantity, the copula, is, signifies 
comprehends. 

Let us now generalize and symbolize this argument, 
transposing the premises, thus : 



S comprehends M. 
M comprehends P. 
.-. S comprehends P. 




ARGUMENTS. 47 

It is to be observed that S, which is the least in 
extensive quantity, is the greatest in comprehensive 
quantity, and that P, which is the greatest in extensive 
quantity, is the least in comprehensive quantity. 

The names, major and minor terms, as defined, are 
significant only in extensive quantity, and even in this 
quantity, not always. The major term, as a matter of 
fact, is frequently less in extension than the minor 
term. They are, therefore, to be regarded as mere 
technical expressions, the major term denoting the 
predicate of the conclusion, and the minor, the 
subject. 

The expressions, major and minor premises, are also 
to be regarded as technical expressions, the major 
premise being the premise containing the major term, 
the minor premise, the premise containing the minor 
term. 

The order of the premises is not essential, though 
the major premise generally stands first. The conclu- 
sion may even be placed before the premises. 



4. Remarks on Mediate Arguments. 

1. The function of an argument is to prove that a 
certain relation exists between two concepts, when 
that relation is not self-evident. 

In mediate arguments, this is accomplished by 
selecting, as the medium of comparison, a third con- 
cept, called, for this reason, the middle concept, with 
which the other concepts are separately compared. 

The separate relations of the extremes to the middle 
prove their relations to each other. 

2. The conclusion must not only be compatible with 






48 



LOGIC. 



the premises, but must be necessitated by them, other- 
wise the argument is a fallacy, thus : 



No P is M. 
No S is M. 
.-. No S is P. 




©^ 



( M )(*) 



Now, though this conclusion may be true, as a fact, 
the premises do not prove it ; for we might have the 
same premises and a contrary conclusion, thus : 

No P is M. 
No S is M. 
.-. All S is 

But this argument, like the other, is invalid, though 
the conclusion is true ; for the premises do not neces- 
sitate the conclusion. 

3. The same relation may have different expressions, 
thus : 




No P is M. 
All S is M. 
.-. No S is P. 




No M is P. 

All S is M. 
.-. No S is P. 



These arguments are identical in thought, as is seen 
by the figures, with an accidental difference of ex- 
pression. 

5. Laws warranting the Conclusions. 

(A), (E), (I), (0). 



To prove (A). 



All M is P. 
All S is M. 
.-. All S is P. 




ARGUMENTS. 



49 



Hence, a universal affirmative conclusion is war- 
ranted, if all of the middle is contained under the 
predicate and all of the subject, under the middle. 



To prove (E). 



f All P is M. 

\ No S is M. 
I /. No S is P. 

f All S is M. 
< No P is M. 
I .-. No S is P. 




Hence, a universal negative conclusion is warranted, 
if all of one extreme, and none of the other, is con- 
tained under the middle. 



To prove (I). 



C* m A 




c All M is P. • 

) All M is S. 
I .-. Some Sis P.. 
c All M is P. ' 
\ Some M is S. 
I :. Some Sis P.. 
r All M is S. ; 
\ Some M is P. 
[.-.Some Sis P.. 

Hence, a particular affirmative conclusion is war- 
ranted, 

1st. If all of the middle is contained under each 
extreme. 

2d. If all of the middle is contained under one ex- 
treme and a part of it, under the other. 

In both cases, the same thing — either all or the same 
part of the middle — is contained under both extremes; 
hence, the extremes must, in part, at least, coincide 
l. 5 



50 



To prove (0).< 




LOGIC. 

No P is M. 

| Some S is M. 

. .-. Some S is not P. 

All P is M. 

Some Sis not M. 
. /. Some Sis not P. 
' All M is S. 
1 Some Mis not P. 
k .'. Some S is not P. 



Hence, a particular negative conclusion is warranted, 

1st. If all of the predicate is excluded from the 
middle and some of the subject is contained under the 
middle. 

2d. If all of the predicate is contained under the 
middle and some of the subject is excluded from the 
middle. 

3d. If all of the middle is contained under the sub- 
ject and some of the middle is excluded from the 
predicate. 

In all these cases, the extremes are so related to the 
middle that some of the subject is excluded from the 
predicate. 

6. Formal Fallacies. 

1. Undistributed Middle. 

It has already been stated that all universals dis- 
tribute the subject, and all negatives the predicate, 
that no particular distributes the subject, and that an 
affirmative may or may not distribute the predicate. 

(A) distributes the predicate in case S is coextensive 
with P, which is the case in definitions, equations, 
identical propositions, etc. 




ARGUMENTS. 51 

(I) distributes the predicate in case S includes P, 
that is, when S is the genus and P the species. 

In all other cases, the predicate of (A) or (I) is un- 
distributed. 

Let us now take an argument with an undistributed 
middle : 

All P is M. 
All S is M. 

.-. All S is P. 

The conclusion does not necessarily follow from the 
premises, though it may be accidentally true, as seen 
in the second diagram above. 

Let now the major premise be a case of coextension, 
then M will be distributed and the argument valid. 

All P is M. 
All S is M. 
.-. All S is P. 

We frequently meet with such reasonings in scien- 
tific works. The following, the conclusiveness of 
which is unquestioned, is a specimen taken from 
geometry : 

Similar polygons are those which are equiangular 
and have their corresponding sides proportional. 

Regular polygons of the same number of sides are 
equiangular and have their corresponding sides pro- 
portional. 

.-. Regular polygons of the same number of sides 
are similar polygons. 

The major premise is a definition of similar poly- 
gons, and, consequently, its subject and predicate are 
coextensive, and the predicate distributed. 




52 



LOGIC. 



The minor premise is a demonstrated proposition. 
The reasoning is valid, and the conclusion true. 
Again, take the following : 

S = M. 
P=M. 

.-. S = P. 

Let the minor premise be a case of coextension. 

All P is M. 
All S is M. 
.-. Some S is P. 



ISMMM) 




Let us take another argument with an undistribute 
middle : 

All P is M. 
Some S is M. 
,\ Some S is P. 

The conclusion is not necessarily true, but may be 
accidentally true, as seen above. 

Let the major premise be a case of coextension. 

aii p is m. /^y<r\ 

Some S is M. ( s ( ) p ) 
.-. Some S is P. VAcJV 

Let the minor premise be a case in which S includes 
M, thus : 

All P is M. 
Some S is M. 
.-. Some S is P. 

Hence, if the middle term is undistributed, no con- 
clusion is warranted ; but it will suffice if the middle 
term is distributed in one of the premises. 







ARGUMENTS. 53 



The fallacy of the undistributed middle consists in 
the fact that but a part of the middle is compared with 
the extremes in the two premises, and it is not certain 
that it is the same part. The extremes, then, are not 
known to be compared with the same thing, and there 
is no warrant for inferring their relations to each 
other. 

2. Illicit Process. 

An illicit process consists in distributing either the 
major or the minor term in the conclusion when it is 
undistributed in its premise, and thus affirms univer- 
sally in the conclusion what is affirmed partially in the 
premises. 

Let us take an argument with an illicit process of 
the major term. 

All M is P. 
No M is S. 
.-. No S is P. 

If the major premise is a case of coextension, tne 
argument is valid, thus : 

All M is P. /-— \ /-— X 

No M is S. 
.-. No S is P. 

Let us now take an argument with an illicit process 
of the minor term. 

All M is P. 

All M is S. 
.-. All S is P. 






54 



LOGIC. 



If the minor premise is a case of coextension, the 
argument is valid, thus : 

All M is P. 
All M is S. 
/. All S is P. 




Let us take another argument with an illicit process 
of the minor term. 






No M is P. 

All M is S. 
.-. Eo S is P. 







If the minor premise is a case of coextension, the 
argument is valid, thus : 

NO M is P. /— X y— N. 

All M is S. p (m s ) 

.-.No sis p. v_y v_y 

3. Particular Premises. 

Except those cases in which an affirmative distrib- 
utes the predicate, particular premises involve either 
an undistributed middle, or an illicit process, and, 
therefore, warrant no conclusion. 

1st. Particular premises with an undistributed middle. 

Some P is M. /""^X^C^X 
Some S is M. ( s ( ) M ( ) p ) 
/.Some Sis P. VJ*L3C^ 

If the subject includes the predicate in either or both 
of the premises, the argument is valid, thus : 

Some P is M. 
Some S is M. 
/. Some S is P. 









ARGUMENTS. 55 

Again, take the following : 

Some M is not P. 

Some S is M. 

/. Some S is not P. 

If the subject of the minor premise is coextensive 
with the predicate, the argument is valid, as in the 
following example : 

Some M is not P. 
Some S is M. ( s 

; /. Some S is not P. 

2d. Particular premises involving an illicit proceed 
of the major term. 

Some M is P. 
Some S is not M. 
.-. Some S is not P. 

If the subject of the major premise includes che 
predicate, the argument is valid. 

Some M is P. 
Some S is not M. 
,\ Some S is not P. 

4. Negative Premises. 

If both premises are negative, no conclusion is war- 
ranted; for the denial of certain relations between 
the middle term and the extremes, warrants neither 
the affirmation nor denial of any relation between the 
extremes : 








No P is M. 
No S is M. 
No S is P. 




56 



LOGIC. 



Take also the following : 

Some M is not P. 

No S is M. ( M (](T)i 

•\ Some S is not P. 




5. An Affirmative Conclusion and One Negative Premise. 

The affirmative premise expresses the agreement, 
in whole or in part, of one of the extremes with the 
middle; and the negative premise, the disagreement 
of the other extreme with the middle ; hence, the ex- 
tremes must disagree with each other, or the con- 
clusion is negative ; hence, an affirmative conclusion 
would be unwarranted, as seen in the following 
examples : 

No M is P. 
All S is M. 
.-. All S is P. 

No M is P. 
Some S is M. 
,\ Some S is P. 




6. A Negative Conclusion from Affirmative Premises. 

Since the premises are affirmative, both extremes 
are affirmed to agree, in whole, or one in whole and 
the other in part with the middle; hence, they must 
agree with each other, or the conclusion is affirmative ; 
hence, a negative conclusion would be unwarranted, 
as in the following examples: 

All M is P. 
All S is M. 
/. No S is P. 




ARGUMENTS. 



57 




All M is P. 

Some S is M. 

.\ Some S is not P. 



7. A Universal Conclusion and a Particular Premise. 

This involves either an undistributed middle or 
illicit process : 

Some M is P. 

All S is M. (/?)} 

.-. All S is P. 

All M is P. 
Some S is M. 
.-. All S is P. 

No M is P. 
Some S is M. 
.\ No S is P. 

Some M is not P. 
All S is M. 
.-. No S is P. 

Some M is not P. 

All M is S. 
.-. No S is P. 




8. Ambiguous Middle. 

Light is contrary to darkness. 

Feathers are light. 

/. Feathers are contrary to darkness. 



7. Rules. 

1. If both premises are affirmative, the conclusion 
|8 affirmative. 



58 LOGIC. 






2. If one premise is affirmative and the other nega- 
tive, the conclusion is negative. 

3. If both premises are negative, there is no conclu- 
sion. 

4. If the conclusion is universal, both premises must 
be universal. 

5. If both premises are particular, there is no con- 
clusion except in case an affirmative distributes its 
predicate. 

6. The middle term must not be ambiguous. 

7. The middle term must be distributed in one of 
the premises. 

8. No term must be distributed in the conclusion 
which is not distributed in one of the premises. 

8. General Laws of the Syllogism. 

r lst. The truth of the premises involves 
the truth of the conclusion. 
1. Positive. I 2d. The falsity of the conclusion in- 
volves the falsity of one of the 
premises. 
Is*. The falsity of the premises does not 
involve the falsity of the conclusion. 
1 2d. The truth of the conclusion does 
not involve the truth of the premises. 

9. Figure. 

1. Definition. 

Figure is the classification of syllogisms according, 
to the position of the middle term with respect to the 
extremes in the premises. 



ARGUMENTS. 59 

Omitting the copula and the signs distinguishing 
the quantity and quality of the propositions, we have 
the following : 

2. Classification. 

In the first figure, the middle term is 
Fig- l-<j S M. the subject of the major premise and 

predicate of the minor. 

In the second figure, the middle term 
Fig. 2.<j s m. is the predicate of both premises. 




In the third figure, the middle term is 
Fig. 3. ^ m g # the subject of both premises. 

In the fourth figure, the middle term 
Fig- 4. ^ ]y[ g # is the predicate of the major premise 
and the subject of the minor. 

For the opinions of Logicians concerning Fig. 4, see Mahan, pp. 121-4 ; 
Hamilton, pp. 285, 302, 626; Coppee, p. 117; Tappan, p. 347; Thomp- 
son, pp. 201-6; Wilson, p. 110; Whately, p. 96. 

10. Mood. 

1. Definition. 

The mood of a syllogism is the arrangement of its 
propositions according to their quantity and quality. 

2. Remark. 

In the following discussion of the valid moods, we 
shall give those only which are universally valid, dis- 
regarding those exceptional cases, mentioned under 
the head of formal fallacies, which become valid in 



60 



LOGIC. 



consequence of the distribution of the predicate of an 
affirmative. 



3. Positive Determination of the Valid Moods. 

1st. If the conclusion is (A) both premises must be 
(A). [Rules 1, 2, 4]. 



AAAisavalidf All M is P. 



mood, since | 



All S is M. 
\ All S is P. 



is a valid argu^ 
ment. 



2d. If the conclusion is (E), one premise must be 
(A), the other (E). [R. 1, 2, 4]. 



AEE^ 



I are valid 

I moods, since 
EAEJ 



All P is M. 
\ No S is M. 
' .-. No S is P. 
(No P is M. 
< All S is M. 
t.\ No Sis P. 






are valid 
arguments. 






8d. If the conclusion is (I), both premises must be 
affirmative, and one, at least, universal. [R. 1, 2, 5]. 



AAI 
All 



) 



are 



valid 



All M is P. 
All M is S. 



moods, since 
IAI J 



L-.SomeSisP., 
(All Mis P. ' 

< Some M is S. 
t.-. Some Sis P.. 
(Some Mis P." 

< All M is S. 

I. -.Some Sis P.- 



! 



are 



valid 
arguments 



ARGUMENTS. 



61 



4th. If the conclusion is (0), one premise must be 
affirmative, the other negative, and one, at least, uni- 
versal. [R. 1, 2, 5]. 



AEO 



EAO 



AOO 



OAO 



EIO 



IEO 



are valid 
moods, <( 
since 



( All P is M. 
J No S is M. 
v .-. Some S is not P. 

!No M is P. 
All M is S. 
.\ SomeS is not P.. 
rAllPis M. 
< Some S is not M. 
l.\ Some Sis not P.. 
f Some M is not P. 
J All M is S. 
( .-. Some S is not P. 
r No M is P. 
] Some S is M. 
( .\ SomeS is not P. 

!Some S is M. 
No P is M. 
.-.Some Sis not P. 



are valid 
arguments. 



4. The Number of Valid Moods. 



One mood proves (A). 
Two moods prove (E). 
Three moods prove (I). 
Six moods prove (0). 



\ 1 + 2 + 3 + 6=12=the 
number of valid moods. 



The mood I E is valid only on the condition that 
the minor premise stand first, otherwise there would 
be an illicit process of the major term. 



62 



LOGia 



5. Negative Determination of the Valid Moods. 



Major 
Prem. 


Minor 
Prem. 


Con. 


Moods. 


Remarks. 


Major 
Prem, 


Minor 
Prem, 


Con. 


Moods. 


Remarks, 






A 


AAA 


Valid. 






A 


EAA 


I. R. 2. 




A 


E 
I 


AAE 
A AI 


Invalid R.l. 

V. 




A 


E 
I 


EAE 
E A I 


V. 

I. R. 2. 









AAO 


I. R. 1. 









EAO 


V. 




A 


AEA 


I. R. 2. 




A 


EEA 


I. R. 3. 




E 


E 
I 


AEE 
AE I 


V. 

I. R. 2. 




E 


E 
I 


E EE 
E E I 


I. R. 3. 
I. R. 3. 









AEO 


V. 









E EO 


I. R. 3. 


A 










E 






























A 


A I A 


I. R. 4. 






A 


E I A 


I. R. 2. 4. 




I 


E 
I 


A I E 
All 


I. R. 1. 4. 
V. 




I 


E 
I 


E I E 
E I I 


I. R. 4. 

I. R. 2. 









A I 


I. R. 1. 









E I 


V. 




A 


AOA 


I. R. 2. 4. 




A 


EOA 


I. R. 3. 4. 







E 


AOE 


I. R. 4. 







E 


EOE 


I. R. 3. 4. 




I 


A I 


I. R. 2. 




I 


E I 


I. R. 3. 









AOO 


V. 









EOO 


I. R. 3. 



The valid moods are determined by first finding the 
invalid ones; that is, those violating one or more of 
the rules. 

The remaining moods are, of course, valid. 



ARGUMENTS. 



63 



Negative Determination Continued. 



Major 
Prem. 


Minor 
Prem. 


Con, 


Moods. 


Remarks, 


Major 
Prem. 


Minor 
Prem. 


Con. 


Moods, 


Remarks, 






A 


I A A 


I. R. 4. 






A 


OAA 


I. R. 2. 4. 






E 


I AE 


I. R 1. 4. 




A 


E 


OAE 


I. R 4. 




A 


I 


I A I 


V. 






I 


AI 


I. R 2. 









I A 


I. R. 1. 









OAO 


V. 




A 


I E A 


I. R 2. 4. 




A 


OEA 


I. R 3. 4. 






E 


I E E 


I. R. 4. 




E 


E 


OEE 


I. R 3. 4. 




E 


I 


I E I 


I. R 2. 






I 


E I 


I. R 3. 









I E 


V. 









OEO 


I. R 3. 


I 









































A 


I I A 


I. R 4. 5. 






A 


I A 


I.R 2. 4. 5. 






E 


I I E 


I.R. 1.4. 5. 




T 


E 


I E 


I. E. 4. 5. 




I 


I 


III 


I. R. 5. 






I 


Oil 


I. R. 2. 5. 









I I 


I. R 1. 5. 









10 


I. R 5. 




A 


I A 


I. R 2. 4. 5. 


A 


OOA 


I. R. 3. 4. 5. 






E 


I E 


I. R 4. 5. 







E 


OOE 


I. R. 3. 4. 5. 







I 


I I 


I. R. 2. 5. 




I 


1 


I. R 3. 5. 









IOO 


I. R. 5. 









000 


I. R. 3. 5. 



The valid moods, as determined by the positive and 
negative methods, are the same. 

In the columns headed " Remarks," E denotes rule; 
V, valid; I, invalid. 



64 LOGIC. 

11. Figure I. -{ s M. } 
is P. J 

1. Valid Moods in Figure I. 

To have an affirmative conclusion, both premises 
must be affirmative. 

The major premise must be universal, otherwise 
the middle term would not be distributed, since it 
is not distributed as the predicate of the affirmative 
minor premise. 

If the minor premise is universal, the conclusion 
may be universal or particular. If the minor prem- 
ise is particular, the conclusion is particular. 
(AAA.) 

.*. jAAL [ are valid affirmative moods in Fig. 1. 
VAIL ) 

To have a negative conclusion, the major premise 
must be negative in order to distribute the predicate 
which is distributed in the conclusion. 

The minor premise must be affirmative, otherwise 
both premises would be negative, and there would be 
no conclusion. 

The major premise must be universal, in order to 
distribute the middle term, since it is not distributed 
as the predicate of the affirmative minor premise. 

If the minor premise is universal, the conclusion 
may be universal or particular. 

If the minor premise is particular, the conclusion is 
particular. 

(EAE.) 

•"• I E A O. I are valid negative moods in Fig. 1. 

'eio. ) 

Discarding A A I and E A as involved in A A A 
and E A E, we have 



ARGUMENTS. 



65 



2. The Doctrine of Figure L 

1st. The middle term is the subject of the major 
premise and predicate of the minor. 

2d. The major premise is universal, and the minor, 
affirmative. 

3d. The conclusion agrees in quality with the major 
premise, and in quantity with the minor. 

4dh. AH forms of conclusion, (A), (E), (I), (0), are 
admissible in Figure 1. 

3. Aristotle's Dictum. 

Whatever is predicated, affirmatively or negatively, 
of any term distributed, may, in like manner, be predi- 
cated of whatever is contained under that term. 

Let the dictum be applied to each of the following 
arguments. 

4. Arguments in Figure I with their Names. 

( All M is P. 
b A r b A r A J All S is M. 

( .-. All S is P. 

( No M is P. 
c E 1 A r E n t J All S is M. 
( .-. No S is P. 

r All M is P. 
d A r 1 1 . . . . ] Some S is M. 
( .-. Some S is P. 




f E r I 



No M is P. 
Some S is M. 
/.Some Sis not P. 



QQQ 



The vowels in these names designate the propositions. 

L. 6 



66 LOGIC. 



12. Figure II. 




1. Valid Moods, in Figure II. 

In order to distribute the middle term, since it is the 
predicate of both premises, one of the premises must 
be negative ; hence, the other premise must be affirma- 
tive, otherwise there would be no conclusion. 

Since one premise is affirmative and the other 
negative, the conclusion is negative, and, therefore, its 
predicate is distributed ; hence, the major premise must 
be universal in order to distribute its subject, which is 
the predicate of the conclusion. 

If the minor premise is universal, the conclusion 
may be universal or particular. 

If the minor premise is particular, the conclusion is 
particular. 

'EAE | give universal nega-- 
AEE J tive conclusions, 
E AO ^ \ and are valid moods 

AEO I give particular nega- 
EIO | tive conclusions, 
lAOOJ 

Discarding the moods E A and A E as involved 
in E A E and A E E, we have, from the four remaining 
moods, 



in Fig. 2. 






ARGUMENTS. 



67 



2. The Doctrine of Figure II. 

1st. The middle term is the predicate of both 
premises. 

2d. One premise is affirmative and the other neg- 
ative. 

Sd. The major premise is universal. 

4th. The conclusion is negative and agrees in quan- 
tity with the minor premise. 

3. Arguments in Figure II with their Names. 



(No Pis M. 
cEs ArE.. J All Sis M. 
I .-.No Sis P. 




0»> 



(-All Pis M. 

cAmEstrEs^No S is M. 
L-.tfoS is P. 



©«) 





(No P is M. 
fEstlnO. J Some Sis M. 

I .*. Some S is not P. 

(All P is M. 
I A k r . . .1 Some S is not M. 
I .-. Some S is not P. 



Aristotle's dictum, given on page sixty-five, does 
not apply directly to any of the four figures except the 
first. 




68 LOGIC. 



(M P.) 
13. Figure III. I M S. \ 

1. Valid Moods in Figure III. 

In order to distribute the middle term, one of the 
premises must be universal. 

If both premises be universal affirmative, and a uni- 
versal affirmative conclusion be drawn, there would be 
an illicit process of the minor term. 

If both premises be universal, the major negative 
'and the minor affirmative, and a universal negative 
conclusion be drawn, there would be an illicit process 
of the minor term. 

If both premises be universal, the major affirmative 
and the minor negative, and a universal negative con- 
clusion be drawn, there would be an illicit process of 
the major term. 

Hence, in Fig. 3, the conclusion must be particular 
If the conclusion is affirmative, both premises must 
be affirmative. 

If the conclusion is negative, the major term is dis- 
tributed; hence, the major premise must be negative, 
and, therefore, the minor premise, affirmative. 



give particular affirm 

ative conclusions, 

' and are valid moods 

give particular nega- | m &' 
tive conclusions, 




ARGUMENTS. 



69 



2. The Doctrine of Figure III. 

1st. The middle term is the subject of both premises. 

2d. One premise is universal and the minor affirma- 
tive. 

3d. The conclusion is particular and agrees in qual- 
ity with the major premise. 

3. Arguments in Figure III with their Names. 



r All M is P. 
dArAptl .\ AllMisS. 

I .-. Some S is P. 

r Some M is P. 
dlsAmls J AllMisS. 

( .\ Some S is P. 

c All M is P. 
d A 1 1 s I . . A Some M is S. 

I .-. Some S is P. 

r No M is P. 
fElAptOnJ AllMisS. 

I .-. Some Sis not P. 

r Some M is not P. 
d k A m } All M is S. 

I .-. Some S is not P. 

r No M is P. 
fErlsO.. J SomeM is S. 

(. .-. Some S is not P. 



(s (®) rj 




70 



LOGIC. 




U. Figure IV. 

1. Valid Moods in Figure IV. 

In order to distribute the middle term, the major 
premise must be negative or the minor, universal. 

If the conclusion is affirmative, both premises must be 
affirmative, and the minor premise universal in order 
to distribute the middle term which is not distributed 
as the predicate of the affirmative major premise. 

The affirmative conclusion must be particular, other- 
wise there would be an illicit process of the minor term. 

If the conclusion is a universal negative, both prem- 
ises must be universal, the minor, negative, in order to 
distribute its predicate which is distributed as the sub- 
ject of the universal conclusion, and, therefore, the 
major premise must be affirmative. 

If the conclusion is a particular negative, the major 
premise must be universal in order to distribute its 
subject which is distributed as the predicate of the 
negative conclusion ; and if the major premise is af- 
firmative, its predicate, which is the middle term, is 
undistributed; hence, the minor premise must, in this 
case, be a universal negative; but if the major prem- 
ise is negative, the minor premise may be either a 
universal or a particular affirmative. 

r AAI ) give affirmative 
conclusions, 



IAI 
AEE 
AEO 
EAO 
EIO 



negative 



^give 

conclusions, 



and are valid moods 
Figure 4. 



Discarding A E as involved in A E E, we have 



ARGUMENTS. 



71 



2. The Doctrine of Figure 1 V. 

1st. The middle term is the predicate of the major 
premise and subject of the minor. 

2d. Either the major premise must be negative or 
the minor, universal. 

3d. If the conclusion is affirmative, both premises 
must be affirmative; the minor, universal, the major, 
universal or particular, and the conclusion particular. 

4:th. If the conclusion is a universal negative, the 
premises must both be universal ; the minor, negative, 
the major, affirmative. 

hth. If the conclusion is a particular negative, the 
major premise must be a universal negative, and the 
minor premise may be either a universal or a particu 
lar affirmative. 

3. Arguments in Figure IV with their Names. 

f All P is M. 
brAmAntlpj All M is S. 

' .-. Some S is P. 



c A. m E n E s 



d ImArls, 



fEsApO 



frEsIsO n. 



f All P is M. 

\ No M is S. 
I .-. No S is P. 

f Some P is M. 
■ < All M is S. 
I .-. Some S is P. 

r No P is M. 
\ All M is S. 
I .\ Some Sis not P. 

f No P is M. 

\ Some M is S. 

t ,\ Some S is notP, 




LOGIC. 



15. Summary of the Names of the Arguments. 






Fig. I. bArbArA, cElArEnt, dArll, 
f E r I 0. 

Fig. II. cEsArE, cAmEstrEs, fEstlnO, 
fAkOrO. 

Fig. III. d A r A p t I, dlsAmls, d A 1 1 s I, 
fElAptOn, dOkAmO, fErlsO. 

Fig. IV. brAmAntlp, cAmEnEs, dlm- 
A r I s, fEsApO, frEsIsOn. 

16. Signification of the Consonants in Fig's II, III, IT. 

1. Consonants denoting Results. 

1st. Initial b denotes reduction to b A r b A r A. 
2d. Initial c denotes reduction tocElArEnt. 
Sd. Initial d denotes reduction to d A r 1 1. 
4th. Initial / denotes reduction to f E r I 0. 

2. Consonants denoting Transformation. 

1st. m denotes that the premises are to be trans- 
posed. 

2d. s denotes that the proposition represented by 
the preceding vowel is to be converted simply. 

Sd. p denotes that the proposition represented by 
the preceding vowel is to be converted by limitation; 
but p in b r A m A n t I p denotes that when the 
reduction is made, a universal conclusion is warranted. 

4th. k denotes that the preceding A is to be changed 
into JE, and the result converted simply, and that the 
preceding is to be changed into I, and the result 
converted simply. 



ARGUMENTS. 



73 



17. Direct Reduction. 



1. Object. 



The object of reduction is to bring arguments in 
Figures II, III, and IV to the test of Aristotle's Dictum, 
which is applicable only to arguments in Figure I. 



2. Fig. II to Fig. I. 



o E s A r E 



cAmEstrEs 



f Es tin 



f AkOr 



No P is M. 

All S is M. 
.-. No S is P. 

All P is M. 
No S is M. 
.-. No S is P. 



^ r No M is P. 

^=cElArEntj All S is M. 
J I .-. No S is P. 



r No P is M. 

\ Some S is M. 

** .\ Some S is not P. 



r All P is M. 

j Some S is not M 

*-.-. SomeS is notP 



}- 
} = 



r No M is S. 
cElArEntj All P is M. 
* .-. No P is S. 



r No M is P. 
f E r I j Some s i s M. 

•*■.*. SomeS is not P. 

C No non-M is P. 
f E r I j g ome s is non-M. 
> .-. Some S is not P. 



3. Fig. Ill to Fig. I 





r All M is P. 


1 


r All M is P. 


dAr AptI 


\ All M is S. 


I =dArII 


■j Some S is M. 




1 .\ Some S is P. 


J 


*■ /. Some S is P. 




r Some M is P. 


) 


r All M is S. 


dlsAmls 


\ All M is S. 


[ =dArIl 


\ Some P is M. 




1 .*. Some S is P. • 


) 


^ .-. Some P is S. 




r All M is P. 


1 


r All M is P. 


d Atlsl 


*j Some M is S. 


I =dArII 


j Some S is M. 




^ .-. Some S is P. 


J 


* .: Some S is P. 



f No M is P. 
fElAptOnj AllMisS. 

*• .*. Some S is not P. 
l. 7 



r No M is P. 
f E r I I Some S is M. 

^ ,\ Some S is not P 



74 LOGIC. 

f Some M is not P. } f All M is S. 

dOkAmO-JAllMisS. Y *= d A r 1 1 j Some non-P is M. 

,* .\ Some non-P is S. 



*■.•. Some Sis not P. 

r No M is P. ^| r No M is P. 

f E r I s j Some M is S. Y = f E r I j Some S is M. 

V\ Some S is not P. * ^ .% Some S is not P 

4. Fig. IV to Fig. I. 

C All P is M. -\ r All M is S. 

brAmAntlpj AllMisS. UbArbArA j A11 P is M . 

^ .-. Some S is P. > *■ .-. All P is S. 

(AllPisM, ^| TNoMisS. 
c Am EnEsj No M is S. V = cElArEntj All P is M. 

I .-. No S is P. J ^ .-. No P is S. 

r Some P is M. \ ( All M is S. 

d I m A r I s j All M is S. [ = d A r 1 1 j Some P is M. 

^ .-. Some S is P. J ^ .% Some P is S. 

r No P is M. \ f No M is P. 

f E s A p j All M is S. J" = f E r I j Some S is M. 

I .-. Some S is not P. * ^ .-. Some S is not P. 

{No P is M. -j t No M is P. 

Some MisS. I = f E r I \ Some S is M. 

,\ Some S is not P. J I .-. Some S is not P 

18. Indirect Reduction. 

1. Notation. 

t = true, /= false, c = contradictory, c' = contrary, 
h = hypothesis. 

2. Fig. II to Fig. I. 

Rule. 

Substitute the contradictory of the conclusion foi 
the minor premise. 

r No P is M. -j r No P is M. 

c E s A r W j All S is M. Y gives f E r I j Some S is P. 

*■ .-. No S is P. J I .-. Some S is not M 



ARGUMENTS. 75 

But is the c of A which is t by h ; :. is/; .\ either 
E or I must be/; but E is t by h ; .\ I is/; but I is the 
c of E'; .-. E' is t. 

(AllPisM. "J fAllPisM. 

?AmEstrE / B \ No S is M. Ygives d A r I V J Some g is p. 

I .\ No S is P. J I .♦. Some S is M. 

But I' is the <? of E which is thy h; .*. I' is/; ,\ either 
A or I must be /; but A is t by h ; .'. I is /; but I is 
the c of E'; .\ E' is t. 



f No P is M. } giveg r No P is M. 

SomeSisM. cE lArE^nt A11SisP ' 

1 .-. Some S is not P. J l .-. No S is M. 



But E' is the c of I which is t by h; ;. E' is/; .-. either 
E or A must be/; but E is t by h; ./. A is /; but A is 
the c of 0; /. is t. 



rAiiPisM. i ives fAl 

fAk0rQ/ j Some S is not M. f b A r b A'r A" | A] 



fAllPisM. ) ffivea (AllPisM. 

All S is P. 
:. Some S is not P. J ^ /. All S is M. 



But A" is the c of which is t by h; .\ A" is /; 
.•. either A or A' must be /; but A is t by h ; ,\ A' is /; 
but A' is the c of 0'; .\ 0' is Z. 

3. Fig. Ill to Fig. I. 

Rule. 

Substitute the contradictory of the conclusion for 
the major premise. 

r All Mis P. 1 giveg r No Sis P. 

dArA'ptl JAllMisS. L E x A , p w n J All M is S. 

1 .-. Some S is P. * l .-. No M is P. 

But E' is the d of A which is t by A; .\ E' is f; 
. either E or A' must be/; but A' is t by h; .\ E is/; 
but E is the c of I ; /.I is £. 



76 LOGIC. 



d I s A m F s 



f Some M is P. } giveg r No S is P. 

All MisS. c ElArE'nt All M is S. 

1 .-. Some S is P. J ^ .-. No M is P. 



But W is the c of I which is t by h ; .-. E' is/; .'. either 
E or A must be /; but A is t by h; .\ E is /; but E 
is the c of I'; .*. I' is t. 

r All I\I is P. ^ r No S is P. 

d A 1 1 s F j So me M is S . J- gives f E r I j Some M is S. 

^ .-. Some S is P. * ^ .-. Some M is not!' 

But is the c of A which is thy h; ,\ is/; .-. either 
E or I must be /; but I is t by h; .*. E is /; but E is 
the c of I'; .\ I' is t. 



f No M is P. ^| iveg r All S is P. 

f E 1 Ap t n j A11 M is s k A'r b A r A" 1 AU M is S ' 

^ • Snmo ft is Tint P * *• • All M ift 



But A" is the & of E which is t by h; .\ A" is /; 
.-. either A' or A must be/; but A is t by h; .\ A' is/; 
but A' is c of ; .\ is t. 

C Some M is not P. \ • eg f All S is P. 

d k A m OM A ii M is S. k A' r b A r A" 1 AU M is S * 

1 .-. Some S is not P. J *> .-. All M is P. 

But A" is the c of which is t by A; .-. A" is /; 
/. either A' or A must be/; but A is t by h; ,\ A' is/ - 
but A' is the c of 0'; .\ 0' is t. 



( No M is P. ~\ f All S is P. 

f E r I s \ Some M is S. \ g ives d A r I F j Some M is S. 

^ • Sattip ft ia nnf P ' ^ • ftr»mf» M is 



But I' is the c of E which is t by h ; .\ V is/; .-. either 
A or I must be /; but I is t by h; .-. A is /; but A is 
the c of ; .-.0 is t. 



ARGUMENTS. 77 

4. Fig. IV to Fig. I. 
Kule. 

Substitute the contradictory of the conclusion for 
the major premise, but in c A m E n E s for the minor. 

C All P is M. -\ gives r No S is P. 

brAmA'ntlpJAHMisS. c E 1 A'r E'n t 1 A11 M is S ' 

1 .-. Some S is P. } ^ .-. No M is P. 

But the converse of E' is the c f of A which is t 
by h; .'. W is /; .-. either E or A' must be /; but A' 
is thy h; ;. E is/; but E is the c of I; .-.I is t. 

C All P is M. } r All P is M. 

c A m E n W s \ No M is S. [ S ives d A r IF j g ome s is P. 

<■ .-. No S is P. ' I .-. Some S is M. 

But the converse of V is the c of E which is t by h, 
/. V is/; .-. either A or I must be /; but A is t by h ; 
.*. I is/; but I is the c of E'; /. W is £. 



r Some P is M. ^| gives f N< 

dlmArFs A11 M is S. cE lArE'nt A! 

V • Snmo R is P > y- • 



( Some P is M. } ffives ( No S is P. 

All M is S. 
- .-. Some S is P. J K .\ No M is P. 



But the converse of E' is the c of I which is t by h ; 
.-. W is /; \ either E or A must be/; but A is t by h ; 
.-. E is/; but E is the c of I'; .\ F is t. 

f No P is M. ^| iyes r All S is P. 

f E s A p j All M is S. k A ' r^A r A" j An M ^ S. 

^ .-. Some S is not P. -* ^ .-. All M is P. 



But the converse of A" is the c of E which is t by h ; 
\ A" is /; .*. either A' or A must be /; but A is t by 
I; .-. A' is /; but A is the c of 0; .\ is t. 



78 LOGIC. 



f No P is M. } f All S is P. 

f r E s I s n j Sorae M is S. f S ives d A r I F j Some M is s# 

^ .-. Some S is not P. > ^ .-. Some M is P. 

But the converse of I' is the c of E which is t by h , 
.-. V is /; .\ either A or I must be /; but I is t by hi 
.-. A is /; but A is the c of ; .*. is t. 

19. Examples. 

Give the figure, mood, and name of the following 
arguments, and those in the II, III, or IV Figure, re- 
duce to the I, both by direct and indirect reduction. 

( Every event has a cause. 

1- "\ The world is an event. 

^ /. The world has a cause. 

f No vicious conduct is praiseworthy. 

2- < All heroic conduct is praiseworthy. 

^ .-. No heroic conduct is vicious. 

( All diligent scholars deserve reward. 

3. < Some boys are diligent scholars. 

v. .-. Some boys deserve reward. 
( All good reasoners are candid. 

4. < Some infidels are not candid. 

^ ,\ Some infidels are not good reasoners. 

{All oaks are trees. 
All trees are vegetables. 
.-. Some vegetables are oaks. 
( Every wicked man is discontented. 
6. < No happy man is discontented. 
^ .-. No happy man is a wicked man. 

{All wits are dreaded. 
All wits are admired. 
/. Some who are admired are dreaded. 



ARGUMENTS. 79 

{Some slaves are not discontented. 
All slaves are wronged. 
.-. Some who are wronged are not discontented. 



r 



9. 



No immoral acts are proper amusements. 

All proper amusements give pleasure. 

.*. Some things that give pleasure are not nn- 
I moral acts. 

( All expedient things are conformable to nature. 

10- \ Nothing conformable to nature is hurtful to society. 

v. /. Nothing hurtful to society is expedient. 

( No one governed by passion is free. 

11- \ All sensualists are governed by passion. 

v .-. No sensualist is free. 

f No just act will result in evil. 

12. < Some association will result in evil. 
^ .-. Some association is not a just act. 

'No impediment to commerce is favorable to na- 
tional prosperity. 

13. ^J Some taxes are impediments to commerce. 
/. Some taxes are not favorable to national pros- 
perity. 

No Science is capable of perfection. 
1 . I All Science is worthy of culture. 

Something worthy of culture is not capable of 
perfection. 
All pride is inconsistent with religion. 
. Some pride is commended by the world. 

Something commended by the world is incon- 
sistent with religion. 

( Some noble characters are not philosophers. 
16. \ All noble characters are worthy of admiration. 
^ ,\ Some worthy of admiration are not philosophers. 






80 LOGIC. 



n.< 



19 A 



'No prejudices are compatible with perfection. 
Some prejudices are innocent. 
,\ Some innocent things are not compatible with 
perfection. 

{Some taxes are oppressive measures. 
All oppressive measures should be repealed. 
.-. Some things which should be repealed are taxes. 

No fallacious argument is a legitimate mode of 

persuasion. 
Some legitimate modes of persuasion fail to gain 

acquiescence. 
.-. Some arguments which fail to gain acquiescence 

are not fallacious. 

20. Hypothetical Syllogisms. 

1. Definition. 

An hypothetical syllogism is an argument whose 
form is determined by the Law of Reason and Con- 
sequent. 

2. Examples. 

T If A has the fever, he is sick. 
Is*. Constructive, j But A has the fever. 
' .-. A is sick. 

!Ii A has the fever, he is sick. 
But A is not sick. 
.-. A has not the fever. 

3. The Propositions of an Hypothetical Syllogism. 

1st. The major premise is an hypothetical proposi- 
tion, definite and affirmative, enouncing the depend- 
ency between a conditioning antecedent and a condi- 






ARGUMENTS. «] 

tioned consequent, but affirming nothing in regard to 
the actual existence of either. 

2d. The minor premise is a categorical proposition 
either affirming the conditioning antecedent or deny- 
ing the conditioned consequent. 

3d. The conclusion is a categorical proposition af- 
firming the consequent, if the antecedent is affirmed 
in the minor premise : or denying the antecedent, if the 
consequent is denied in the minor premise. 

4. Laws. 
fa. Affirming the antecedent affirms the 

L Positive. J u ^ consequent. 

b. Denying the consequent denies the 
< antecedent. 

( a. Denying the antecedent does not 

2d. Negative. J deny the consequent. 

b. Affirming the consequent does not 
affirm the antecedent. 
If the predicate of the antecedent is co-extensive 
with the predicate of the consequent, the negative 
laws become positive. 

5. Categorical and Hypothetical Syllogisms Compared. 
Though it be true that an hypothetical syllogism 
has an hypothetical proposition for its major premise, 
yet it does not follow that every syllogism which has 
an hypothetical major premise is an hypothetical syl- 
logism. Thus, take the following : 

If the Scriptures came from God, they are entitled 
to our faith. 

If they are not an imposture, they came from God. 

If, therefore, they are not an imposture, they are en- 
titled to our faith. 





g2 LOGIC. 

6. Reduction of Hypothetical Syllogisms to Categorical. 

1st. If the major premise contains three terms, one 
being a middle, thus : 

If A is B, AisC. 
But A is B. 
/. A is C. 

2d. If the major premise contains four terms, thus : 

-BisD. 

If A is B, C is D. 

But A is B. 
.-. C is D. 



21. Disjunctive Syllogisms. 

1. Definition. 
A disjunctive syllogism is an argument whose form 

is determined by the Law of Contradictories. 

2. Examples. 

( Plato is either learned or unlearned 
1st Affirmative. < But Plato is learned. 

I \ Plato is not unlearned. 

( The patient will live or die. 
2d. Negative. < He will not live. 
I.-. He will die. 

3. The Propositions of a Disjunctive Syllogism. 
1st. The major premise is a disjunctive proposition 
universal and affirmative, having the opposition, 



ARGUMENTS. 83 

a. Contrary, as S is either P or Q, determined a posteri- 
ori, and thus brought under the Law of Contradictories. 

r a. In the copula, as S either is or 

>b. Contradictory. \ ' . . 

p. hi the terms, as b is either P or 
V non-P. 

2d. The minor premise is a categorical proposition, 
universal or particular, affirmative or negative, remov- 
ing the disjunction. 

3d. The conclusion is a categorical proposition, 
agreeing in quantity, but disagreeing in quality, with 
the minor premise. 

4. Disjunctive Syllogisms having two Disjunctive Members. 

T S is either P or Q. 
1st. Affirmative. < But S is P. 
^.\ S is not Q. 

(Sis either P or Q. 
2d. Negative. < But S is not Q. 






V/.SisP. 

5. Laws. 

1st. Affirming either alternative denies the other. 
2d. Denying either alternative affirms the other. 



6. Disjunctive Syllogisms having more than two Disjunc- 
tive Members. 

f A is either B, C, D, or E. 
A But A is B. 
1st. Affirmative. \ ^ •*• A is neither C, D, nor E. 
(A is either B, C, D, or E. 
*6- i But A is either B or C. 
^ .-. A is neither D nor E. 



84 LOGIC. 



2d. Negative. 



f A is either B, C, D, or E. 
• < But A is neither B, C, nor D. 
L\ AisE. 

f A is either B, 0, D, or E. 
\ But A is neither B nor C. 
v .-. A is either D or E. 

7. Laws. 

1st. Affirming a part of the disjunctives, determin- 
ately or indeterminately, in the minor premise, denies 
all the others in the conclusion. 

2d. Denying a part of the disjunctives in the minor 
premise, affirms the rest, in the conclusion, determin- 
ately or indeterminately, according as one or more 
remain. 

8. Categorical and Disjunctive Syllogisms Compared. 

Though it be true that a disjunctive syllogism has a 
disjunctive major premise, it does not follow that every 
syllogism with a disjunctive major premise is a dis- 
junctive syllogism. 

Take the following: 

B is either C or D. 

AisB. 

/. A is either C or D. 

This syllogism is not disjunctive, but categorical, 
since its form is determined, not by the Law of Con- 
tradictories, but by the Law of Identity. 

22. Dilemmatic Syllogisms. 

1. Definition. 

A dilemmatic syllogism is a syllogism having a 
hypothetical major premise and a disjunctive minor. 



ARGUMENTS. 85 

2. Forms. 

( If A is B, X is Y. 

1st. < If C is D, X is Y. 

<-If EisF, Xis Y. 

( A is B, or "J 
a. Either -j C is D, or V /• X is Y. 

**< lEisF - J f A isB.no,- 

6. X is not Y. /. Neither -< Q is D, nor 

^E is F. 

(If A is B, CisD. 
2i. < If AisB, EisE. 

I If AisB,GisH. 

( C is D, and 
a. A is B. .-. < E is E, and 
But<! UisIL 

Either |CisnotD,or| 

< E is not F, or > .: A is not B. 
<-GisnotH. J 

(If AisB, GisH. 
Bd. < If is D, I is E. 
I- If EisF, Lis M. 

( A is B, or ") f G is II, or 

a. Either <( C is D, or > /.Either «j I is K, or 
LEisF. > <- L isM. 

| (GisnotH,or] f A is not B, or 

U Either \ I is not B, or > .'.Either -j C is not D, or 
'-Lis not M. ) ^EisnotF. 

3. Remark. 

The forms, Is*., b., and 2d., a., are not, strictly, 
dilemmatic syllogisms, since the minor premise is not 
disjunctive. 



86 LOGIC. 

23. Enthymemes. 

1. Definition. 

An enthymeme is a syllogism with one proposition 
suppressed. 

It differs from the ordinary syllogism, not in thought, 
but in enouncement. 

2. Etymology. 

The word enthymeme is from ^^u^^a, kv and &ufi6q y in 
the mind. 

3. Examples of Enthymemes. 

1st. With a suppressed f Caesar is a man. 
major premise. \ .-. Caesar is mortal. 

2d. With a suppressed J All men are mortal, 
minor premise. I .-. Caesar is mortal. 

3d. With a suppressed ( All men are mortal, 
conclusion. ( Caesar is a man. 

Each of these enthymemes is equivalent to 

f All men are mortal. 
The complete syllogism.^ Csesar is a man. 

^.\ Caesar is mortal. 
2i. Prosyllogism and Episyllogism. 

1. Definitions. 

1st. A prosyllogism is an argument whose conclusion 
is one of the premises of another argument. 



ARGUMENTS. 



87 



2d. An episyllogism is an argument one of whose 
premises is the conclusion of another argument. 



Prosyllogism. 



2. Example. 

(All Bis a 

< All A is B. 
<• .-. All A is 0. 



Main Syllogism. 
Episyllogism. 



A11C is D. 
All A is C. 
-.-. All AisD. 

AllDisE. 
All A is D. 
-.\ All A is E. 



25. Sorites. 

1. Definition. 

The Sorites or Chain Syllogism is a compound ar- 
gument. 

2. Forms. 

1st. When the predicate of each premise is the sub- 
ject of the next. 



a. Affirmative. 



6. Negative. 



f All A is B. 

All B is C. 

All C is D. 
v.-. All AisD. 



f All A is B. 
J All B is C. 
| No C is D. 
L\ No A is D. 




88 



LOGIC. 



Id. When the subject of each premise is the predi- 
cate of the next. 



a. Affirmative. 



b. Negative. 



f All Bis A. 
All C is B. 
All D is C. 
.-. All D is A. 

No B is A. 
All C is B. 
All D is C. 
.-. No D is A. 






3d. "When the first and second forms are combined. 

r All A is B. 
All B is C. 
No G is D. 
All E is D. 
All F is E. 
All G is F. 
.-. No A is G. 

r All A is B. 
All B is C. 
All C is D. 
No E is D. 
All F is E. 
All G is F. 
.-. No A is G. 

3. Laws. 

111. The first premise in the first form, the last in the 
eeco/i J, and the first or last in the third are the only 
ones 'nat can be particular; and the subject of the par- 
ticular premise will be the subject of the conclusion. 





ARGUMENTS. 



89 



2d. Only one premise can be negative — the last in 
the first form and the first in the second form. 

3d. In the third form, one premise must be nega- 
tive — the last in the first series, or the first in the 
second series. 

4. Expansion of the Sorites. 



ut 



fAll A is B. 
All B is C. 
All C is D. 
.-. All A is D. 



r All A is B. 
j All B is C. 
I .-. All A is C. 

All C is D. 

All A is C. 

.-. All A is D. 



[ { 




r The mind is a thinking substance. 
A thinking substance is a spirit. 
A spirit has no composition of parts. 
2d. Expand ^ That which has no composition of parts 
is indissoluble. 
That which is indissoluble is immortal. 
.-. The mind is immortal. 



26. The Epichirema. 

1. Definition. 
The Epichirema is an argument in which the 

reasons for the premises are stated in connection with 
them. 

2. Etymology. 

The word epichirema is derived from lni%e(pT)iia, from 
Ini and %e(p, and literally signifies to lay hands upon. 

3. Examples. 

( A is B, for A is C and C is B. 
P*. < D is A, for D is E and E is A. 
s.-.DisB. 

L. 8 






2d. 



90 LOGIC. 

'All true patriots are friends to religion, because 
religion is the basis of national prosperity. 

Some great statesmen are not friends to religion, 
because their lives are not in accordance with 
its precepts. 

.-. Some great statesmen are not true patriots. 

27. The Unfigured Syllogism. 

1. Definition. 

The Unflgured Syllogism is an argument in which 
the terms of the propositions do not sustain to each 
other the relation of subject and predicate. 

2. Examples. 

( A and B always coexist. 
1st. Positive. < B and C always coexist. 

v .-. A and C always coexist. 

( A and B always coexist. 
2d. Negative, j B and C never coexist. 

( .-. A and C never coexist/ 

3. Dams. 

1st. As far as two terms agree with a third, so fax 
they agree with each other. 

2d. As far as one term agrees and another disagrees 
with a third, so far they disagree with each other. 

28. The Reductio ad Absurdum. 

1. Definitions. 

1st. An axiom is a self-evident truth. 

2d. An absurdity is a self-evident falsity. 

For every axiom there is a corresponding absurd 



ARGUMENTS. 91 

ity, and for every absurdity there is a corresponding 
axiom, and the two are contradictories. 

3d. The reductio ad absurdum is an argument in 
which a proposition is proved true by showing that 
the supposition that it is false, or, which is the same 
thing, that its contradictory is true, involves an ab- 
surdity. 

2. Principles. 

1st. All truths harmonize. 

2d. If the premises of an argument are true, and 
the reasoning logical, the conclusion is true. 

Sd. If the conclusion of an argument is false, and 
one of the premises true, and the reasoning logical, 
the other premise is false. 

4th If a proposition is true, its contradictory is 
false, and if a proposition is false, its contradictory is 
true. 

3. Application. 

To prove a given proposition true by the reductio 
ad absurdum method, we assume it false; that is, we 
assume its contradictory true. 

We then combine this assumed proposition, with a 
proposition known to be true, in a logical argument 
which gives a false conclusion, false either because it 
is the contradictory of an axiom, and hence absurd, or 
because it is the contradictory of an established propo* 
sition, hence involving the absurdity that one truth 
contradicts another. 

Since the conclusion is false, and one of the premises 

true, and the reasoning logical, the other premise which 

is the assumed proposition, the contradictory of the 

iven proposition, is false ; and if false, its contradie- 




92 LOGIC. 

tory, or the given proposition is true, and is, hence, 
demonstrated. 

4. Example. 

Let it be required to prove the following propo- 
sition : 

If a straight line meet two other straight lines at a com- 
mon point, making the sum of the two contiguous angles 
equal to two right angles, the two lines which are met will 
form one and the same straight line. 

Let D B meet A B and C B I 

at their common point B,J 

making A B D + D B C=:two 

right angles, then will A B 

and B C form one and the 

same right line. 

For, if not, suppose AB and any other line thanj 
B C, as B E, to form the same straight line. 

Then will ABD+DBE= two right angles. 

But, by hypothesis, ABD + DBC = two right 
angles. 

Hence, ABD + DBC = ABD + DBE. 

Subtracting ABD from each of these equals, we 
have DBC = DBE. 

That is, a part is equal to the whole, which is absurd. 

Hence, the supposition that A B and B C do not 
form one and the same straight line, or, which is the 
same, that A B and some other line than B C, as B E, 
form one and the same straight line, involves the ab- 
surdity that a part is equal to the whole; and, there- 
fore, this supposition is false. 

But if it is false that A B and B C do not form one 
and the same straight line, it is true that A B and B C 
do form one and the same straight line. 



ARGUMENTS. 93 



29. The Exhaustiro Method. 

1. Definition. 

The exhaustive method of demonstration is the 
method of proving that a certain relation exists be- 
tween two terms by considering all possible relations, 
one of which must be true, and by showing that all 
except one is false, because involving absurdities, and, 
hence, that the remaining case must be true. 

2. Compared with the Reductio ad Absurdurn. 

In the reductio ad absurdurn method, two cases only 
are considered — the given proposition and its contra* 
dictory. 

In the exhaustive method, several cases are pos- 
sible. 

But since all of these' cases, except one, are shown 
to be impossible by the reductio ad absurdurn method, 
and, hence, are excluded, and since the cases excluded, 
taken together, and the remaining case may be re- 
garded as contradictories, the exhaustive method may 
be considered as an extension of the reductio ad ab- 
surdurn. 

3. Example. 

Two triangles which are mutually equilateral are mutu- 
ally equiangular. 

In the triangles 
ABCandDEF, 
let A B = D E, 
AC = DF, and 
BC = EF, then will the angle A = D, B = E and 
C = F. 




94 



LOGIC. 



For, taking the angles A and ~D, there are three and 
only three cases possible, as follow : 

A>D, A<D,or A = D. 

If A > D, BOEF, which is contrary to the hy 
pothesis ; .-. A is not > D. 

K A < D, BC<EF, which is contrary to the hy- 
pothesis ; .-. A is not < D. 

Hence, since neither A > D nor A < D, A = D. 

In a similar way it can be proved that B = E anc 
C = F. 

30. Hamilton's Notation, and Classification of 
Propositions. 



1. Notation. 




< r : = all in affirmative, and any in neg 
ative propositions. 
, = some. 

^— I = is not. 

-:P is read All S is all P. 

-: P is read Any S is not any P. 

■ , P is read Some S is not Some P. 



2. Proposi- 
tions. 



r lst. Affirm- 
ative. 



fToto-total, (U)S: 
Toto-partial, (A) S: 
Parti-total, (Y) S,i 
Parti-partial, (I) S , 
fToto-total, (E) S: 
2d. Fega- I Toto-partial, (y) S : i 
tive. Parti-total, (0) S , i 

LParti-partial, (o>) S,i 



-:P 
•,F 
•:P 

-,P 
•:P 

•:P 



ARGUMENTS. 



95 



3. The Converse of < 



f(U) S:»— :P = (U) P:' 

(A) S:^-,P = (Y)P,. 

(Y) S,^-:P = (A)P:. 

(I) S,—- ,P = (I) P,' 

(E) S:-4-:P = (E) P:i 

ft) S:^m-,P= (O)P,. 

(0) S,^-:P= 0?) P:« 

[(*) S,^-,P= (») p,. 



4. .Laws o/ Validity. 



-:S. 

•:S. 

-,s. 
■ ,s. 

■:S. 

■ :S. 

,S. 

-,S. 



(U) S : ■>■ — : P. If S and P are coextensive. 



(A) S : m~- , P. If S is subordinate to P. 



(Y) S , — — : P. If P is subordinate to S. 





,© 




(I) S , »— , P. If S and P intersect. 



(E) S :*+-: P. If Sis excluded from P. ( s )( r 



00 S : ■«-« — , P. If S is subordinate to P. 






(O) S , ■h- : P. If P is subordinate to S. L(7\ 



(w) S , »•»— , P. If S and P intersect. 




s p 



96 LOGIC. 



5. Opposition of Judgments. 
u E 






1st. Contradictory opposition, existing between (E) 
and (I). 

a. Law. — Both can not be true nor both false. 

r a. One must be true and the other false 
fi. The truth of either implies the falsity 

of the other. 
r . The falsity of either implies the truth 
of the other. 



b. Corollaries. < 



c. Scholium. — In the classification of categorical propo- 
sitions before given, it was stated that (A) and (0) are 
contradictories, as well as (E) and (I). In the present 
classification, the falsity of (0) does not necessarily im- 
ply the truth of (A), for (U) may be true. The differ- 
ence is owing to this, that in the former classification 
(A) includes both (IT) and (A) of the present classifi- 
cation ; hence, the falsity of (0) implies the truth of 
either (U) or (A). 

2d. Contrary opposition, existing between (U) and 
(E), (U) and (?), (U) and (0), (A) and (E), (A) and (0), 
(Y) and (E), (Y) and (??), one affirmative, the other 
negative. 

a. Law. — Both can not be true, but both may be false. 



ARGUMENTS. 



97 



6. Corollaries. < 



'a. The truth of either implies the falsity 

of the other. 
/?. The falsity of neither implies the 

truth of the other. 



3d. Inconsistent opposition, existing between (IT) 
and (A), (U) and (Y), (A) and (Y), both affirmative. 

a. Law. — Both can not be true, but both may be false. 



6. Corollaries. 



a. The truth of either implies the falsity 

of the other. 
/?. The falsity of neither implies the 

truth of the other. 



Laws. 



4th. Subaltern opposition, existing between (U) and 
(I), (A) and (I), (Y) and (I), (E) and (?), (E) and (0), 
, (E) and («/), (rj) and (V) 5 (0) and (V), both affirmative 
or both negative. 

a. The truth of the superior implies the 
truth of the inferior. 

ft. The falsity of the inferior implies the 
falsity of the superior. 

a. The falsity of the superior does not 
imply the falsity of the inferior. 

/?. The truth of the inferior does not im- 
ply the truth of the superior. 

bth. Subcontrary opposition, between (I) and (0). 
a. Law. — Both can not be false, but both may be true. 

r a. The falsity of either implies the truth 

of the other. 
/?. The truth of neither implies the fals- 
ity of the other. 



b. Corollaries. 



c. Scholium. — The truth of (Y) implies the truth of (O). 

L. 9 



98 LOGIC. 

6. Thompson's Criticism on (??) and (a>). 

"Why have we ventured, in accordance with the 
practice, it is believed, of all, logicians, to exclude these 
two, [(??) and (o>)] ? 

" The answer is, that while Sir William Hamilton 
gives a table of all conceivable cases of negative predi- 
cation, other logicians have only admitted actual cases. 

"It is not inconceivable that a man should say, 'No 
birds are some animals' (the yj of the Table), and yet 
such a judgment is never actually made, because it 
has the semblance only and not the power of a denial. 
True though it is, it does not prevent our making 
another judgment of the affirmative kind, from the 
same terms; and 'All birds are animals' is also true. 

" Though such a negative judgment is conceivable, it 
is useless; and feeling this, men in their daily conver- 
sation, as well as logicians in their treatises, have pro- 
scribed it. 

"But the fruitlessness of a negative judgment, 
where both terms are particular, is even more mani- 
fest; for, 'Some X is not some Y' is true, whatever 
terms X and Y stand for, and therefore the judgment, 
as presupposed in every case, is not worth the trouble 
of forming in any particular one. 

" Thus, if I define the composition of common salt by 
saying 'Common salt is chloride of sodium,' I can not 
prevent another saying that ' Some common salt is net 
some chloride of sodium,' because he may mean that 
the common salt in this salt-cellar is not the chloride 
of sodium in that. A judgment of this kind is spurious 
upon two grounds ; it denies nothing, because it does 
not prevent any of the modes of affirmation ; it decides 
nothing, inasmuch as its truth is presupposed with 

J 



ARGUMENTS. 99 

reference to any pair of conceptions whatever. In a 
list of conceivable modes of predication, these two, 
[(??) and (w)] ? are entitled to a place." 

7. Hamilton's Reply. 

" The thorough-going quantification of the predi- 
cate (on demand) in its appliance to negative proposi- 
tions, is not only allowable, is not only systematic, is 
not only useful, it is even indispensable. For to speak 
of its very weakest form, that which I call parti- 
partial negation, ' Some is not some ;' this (besides its 
own uses) is the form which we naturally employ in 
dividing a whole of any kind into parts : ' Some A is 
not some A.' And is this form — that too inconsist- 
ently — to be excluded from logic? 

" But again (to prove both the obnoxious propositions 
summarily and at once) — what objection, apart from 
the arbitrary laws of our present logical system, can be 
taken to the following syllogism ? 

All man is some animal. 

Any man is not (no man is) some animal. 

Therefore some animal is not some animal. 

"Vary this syllogism of the third figure to any 
other; it will always be legitimate by nature, if ille- 
gitimate to unnatural art. Taking it, however, as it 
is : the negative minor premise, with its particular 
predicate, offends logical prejudice. But it is a propo- 
sition irrecusable; both as true in itself, and as even 
practically necessary. 

" Its converse, again, is technically allowed ; and 
no proposition can be right of whicri the converse 
is wrong. For to say (as has been said from Aris- 
totle downward) that a particular negative propo- 



100 LOGIC. 

sition is inconvertible, this is merely to confess that 
the rules of Logicians are inadequate to the truth of 
logic, and the realities of nature. But this inade- 
quacy is relieved by an unexclusive quantification of 
the predicate. 

"A to to-partial negative can not, therefore, be re- 
fused. But if the premises are correct, so likewise 
must be the conclusion. This, however, is the doubly 
obnoxious form of a parti-partial negative : 

1 Some animal (man) is not some animal (say, brute).' 

"Nothing, it may be observed, is more easy than to 
misapply a form ; nothing more easy than to use a 
weaker, when we are entitled to use a stronger propo- 
sition. But from the special and factitious absurdity 
thus emerging, to infer the general and natural ab- 
surdity of the propositional form itself, this is, cer- 
tainly, not a logical procedure." 

8. De Morgan's Criticism on (w). 

" The proposition, ' Some X's are not some Y's, 9 has 
no fundamental proposition which denies it. . . . It is 
what I have called a spurious proposition, as long as 
either of its names applies to more than one instance. 
And the denial of it is as follows : ' There is but one X, 
and but one Y, and X is Y.' " 



? 



9. Hamilton's Reply. 

"Here, also, Mr. De Morgan wholly misunderstands 
the nature and purport of the form which he professes 
to criticise. He calls it ' a spurious proposition.' .... 
But in no relation can it ever logically be denom- 
inated ' spurious.' 



ARGUMENTS. 101 

"For why? "Whatever is operative in thought, 
must be taken into account, and, consequently, be 
overtly expressible in logic ; for logic must be, as it 
professes to be, an unexclusive reflex of thought, and 
not merely an arbitrary selection — a series of elegant 
extracts, out of the forms of thinking. 

" What then is the function which this form is pecul- 
iarly — is, indeed, alone competent to perform? A 
parti-partial negative is the proposition in which, and 
in which exclusively, we declare a whole of any kind 
to be divisible. ' Some A is not some A ' — this is the 
judgment of divisibility and of division." 

31, Hamilton's Scheme of Figured Syllogisms. 

1. Explanatory Remarks. 

1st. M denotes the middle term, and C and r, in the 
Latin and Greek alphabets, denote the extremes. 

2d. T. B. denotes total balance, both in propositions 
and in terms. 

3d. P. B. denotes partial balance. 

\th. T. U. denotes total unbalance. 

hth. In extension, the broad end of the copula, ■*- — , 
denotes the subject, but in comprehension, the reverse. 

6th. In Fig.'s II and III, the double conclusion is 
denoted by the double copula. 

7th. n- — y~ / denotes that if the premises be con- 
verted, the mood is the same. 

8th. ^>-<c^L denotes that if the premises be con- 
verted, the moods between which it is placed are con- 
vertible into each other. 

9th. The quantity of the terms of the conclusion are 
supposed to be the same as in the premises unless 
otherwise marked. 



102 



LOGIC. 



2. Figure I. 




ARGUMENTS. 



103 



3. Figure II 



Affirmative. 



Negative. 



1. C»— :M: — — :r 



C:pM— :M:- 

h— 



T. B.i 



L 2. C ■- :M:-— *,r 






3. C,^-:M,— *:r 



P. BJ 



4. C:»^-,M:- 



r 5. C,»- :M, 



6. C,^— ,M:- 



7. C:»— :M:- 



T. U.J 





C:i 



,M: 



C:^H— :M, 

4- 



12. C,^—,M:— «s:r 



-,M: — -*:r 

— i m 






C,i 



-:M:- 
H- 




l —a 



C:» — :M,- 



i : r 



C:i 



C: s 



,M:- 

— +- 



:M, 



C,*— ,M:- 



,r 



v.t 



104 



LOGIC. 



T. B. 



L\ B. . 



T. U.1 



4. Figure III. 



Affirmative. 




Negative. 







■Ml \ 



C:— *:M: 



C, — -*:M:i 
H 






H- 



C:— 






c, 






i \-—*0l 



C:- 









C,-^,M:i 



ARGUMENTS. 



105 



32. Indeterminateness of Language. 

One of the greatest obstacles to be surmounted in 
the development of the science of Logic is the indef- 
inite character of language. Thus, take the four 
classes of propositions, generally recognized : 



(A). All S is P. 





(E). No S is P. 






(I). Some S is P. f s A pY Qp) (sQ 




s P 



(0). Some Sis not P. (s A p j f ) ( s ) ( p ) 

(E) is the only proposition free from ambiguity. 

Hamilton's negative propositions, except (E), are 
not free from ambiguity, as shown thus : 






(77). Any S is not some P. 






o 



(0). Some S is not any P. 







106 



LOGIC. 

(uj). Some S is not some P 










©?<@)(^j 



33. Positive Propositions. 




(G). S is coextensive with P. 



(E). S is excluded from P. 



(S). S is subordinate to P. 



(I). S intersects P. 



These four propositions express all possible relations 
of two concepts in extensive quantity. 

(A) and (Y), in Hamilton's scheme, both express the 
relation of subordination. In (A), the subordinate 
concept is the subject, but in (Y), the subordinate con- 
cept is the predicate; but since we can, if we choose, 
always take the subordinate concept for the subject, 
we shall treat (A) and (Y) as one. 

Let the initial letters, (C), (E), (S), and (I), respect- 
ively, denote the relations of coextension, exclusion, 
subordination, and intersection. 

The propositions, (>/), (0), (V), as already shown, are 
indeterminate, being compatible with two or more of 
the above relations, and, therefore, indicate either in- 
adequate knowledge or inadequate expression. 



ARGUMENTS. 



107 






34. Positive Syllogisms. 

1. To prove (C). 

M is coextensive with P. 
M is coextensive with S. 
/. S is coextensive with P. 

Hence, the relation of coextension is warranted in 
the conclusion, if the middle is coextensive with each 
extreme. 

2. To prove (E). 

fP is subordinate to M. 
1st- \ S is excluded from M. 
^.\ S is excluded from P. 

|| S is subordinate to M. 
-J- \ P is excluded from M. 
^ .*. S is excluded from P. 
f P is coextensive with M. 
&d. < S is excluded from M. 
^.\ S is excluded from P. 

r S is coextensive with M. 
4th. < P is excluded from M. 
^.\ S is excluded from P. 







Hence, the relation of exclusion is warranted in the 
conclusion, if one extreme is either subordinate to, or 
coextensive with, the middle, and the other extreme 
is excluded from the middle. 



1st.- 



3. To prove (S). 

M is subordinate to P. 
S is subordinate to M. 
.*. S is subordinate to P. 





108 LOGIC. 

( M is coextensive with P. 
2^- ^ S is subordinate to M. 
^ .*. S is subordinate to P. 

f M is subordinate to P. 
3^. < S is coextensive with M. 
^ .-. S is subordinate to P. 

Hence, the relation of subordination is warranted in 
the conclusion, 

a. If the middle is subordinate to one extreme, and 
the other extreme is subordinate to the middle. 

b. If the middle is coextensive with one extreme, and 
the other extreme is subordinate to the middle. 

c. If the middle is subordinate to one extreme, and 
the other extreme is coextensive with the middle. 

4. To prove (I). 

T M is coextensive with P. 
Is*. < M intersects S. 
^.\ S intersects P. 

T M is coextensive with S. 
2d. < M intersects P. 
^.\ S intersects P. 








Hence, the relation of intersection is warranted in 
the conclusion, if the middle term is coextensive with 
one extreme and intersects the other. 

The figure, mood, and order of the premises are 
mere accidents. Thus, the last S3dlogism is identical 
m thought with the following, though the expression 
is changed : 

M intersects P. 

S is coextensive with M. f ( ) v 

.*. S intersects P. 







ARGUMENTS. 109 

3. INDUCTION. 
1. Definition. 



Induction is the process of establishing general 
propositions from particular cases. 

2. Classification. 

1st. Formal induction, in which the inference is ne- 
cessitated by the Laws of Thought. Formal induction 
is of two kinds : 

a. Logical induction, in which we reason from all 
the parts discretively to the whole collectively. 

b. Mathematical induction, in which we prove a par- 
ticular case, and then that if any case is true, the next 
ease is true, and the next, and so on, indefinitely. 

2d. Real induction, in which we infer that what is 
true of the parts examined is true of the whole. 

3. Position of Induction in Logic. 






Formal induction belongs to Pure Logic. 
Real induction belongs to Modified Logic. 

L Logical Induction. 

1. Definition. 



Logical induction is the process of reasoning from 
all the parts to the whole. 

2. Examples. 

1st. Inductive syllogisms in extension. 



110 LOGIC. 

( a, b, c, are contained under P. 

a - i a, b, c, constitute S. 

^.\ S is contained under P. 

{a, b, c, constitute M. 
S is contained under M. 
.-. S is contained under a, b, c. 
2d. Inductive syllogisms in comprehension. 

{S comprehends a, 6, c. 
a, b y c, constitute P. 
.-. S comprehends P. 
{a, 6, c, constitute M. 
M comprehends P. 
.-. a, 6, c y comprehend P. 

3. Law. 

What belongs, or does not beiong, to all the con- 
stituent parts, belongs, or does not belong, to the con- 
stituted whole. 

5. Mathematical Induction. 

1. Definition. 

Mathematical induction is the process of proving a 
general proposition, by showing that it holds true for 
one or more of the first consecutive cases, and then 
that if it holds for any case, it holds for the next case. 

2. Example. 

The difference of the same powers of two quantities is 
divisible by the difference of those quantities. 



Now 



(a - 


_6)-«-(a — 6) = 1. 


(a 2 - 


_ fj 3 ) -r- (a — b) = a + b. 


(a 3 - 


- 6 s ) ■+■ (a — 6) = a" i a b + b 2 



ARGUMENTS. Ill 






Let us now divide a n — b n by a — b. 



a n — b n 

a n — a n ~ 



a — b 



b — b n -=b (a 11 ' 1 — b n ~ x ). 



Now, it is evident that if a n ~ l — 6 W-1 , which is a factor 
of the remainder, is divisible by a — 6, the whole re- 
mainder, and consequently the dividend, will be divis- 
ible by a — b; that is, if a n ~ l — b n ~ l is divisible by a — 6, 
then a n — b n will be divisible by a — b; hence, If the 
difference of two 'powers of the same degree is divisible 
by the difference of the quantities, the difference of the pow- 
ers one degree greater will be divisible by the difference of 
the quantities. But it has already been found that the 
difference of the powers of the same degree, up to the 
third power, is divisible by the difference of the quan- 
tities, hence, the difference of the fourth powers is 
divisible by the difference of the quantities, and if the 
fourth, then the fifth, and so on, to any degree. 






6. Day's Theory of Induction. 



Let W be a whole of which P is a part and C its 
complementary part, and A an attribute of P so far 
as P is W. 

Then, what is true of P so far as P is W, is true of 
C ; that is, 

Whatever is true of any part of a whole, so far as it 
is a part of that whole, is true of the complementary 
part, thus : 

»P is A so far as P is W". 
C is the complementary of P. 
.-. C is A. 



112 LOGIC. 

Objection to this Theory. 

That P is A so far as P is W, must mean that P is 
A, not because A is characteristic of P, but because A 
is common to all W; that is, because every W is A. 
But this is the very thing to be established by induction. 
Then the so-called induction is resolved into the de- 
ductive syllogism : 

Every W is A. 
Cis W. 
.-. C is A. 

If every W is not A, it does not follow that C is A, 
though P is A. 

It will not do to reason, Sheep have split hoofs; all 
other animals taken together are complementary of 
sheep ; /. all other animals have split hoofs. 

But it may be said that the proposition does not 
comply with the formula, because it was not said, 
Sheep, so far as animals, have split hoofs. But this 
could not be affirmed unless it were known that all 
animals have split hoofs. It is thus evident that there 
is no demonstrative reasoning from part to comple- 
mentary part, except through the whole. The reason- 
ing then becomes deductive. We can indeed reason 
from part to complementary part, without passing 
through the whole ; but the conclusion is then only 
probable, and the induction real not formal. 

7. Whately's Theory of Induction. 

1. Signification of Induction. 

"Induction means properly, not the inferring of the 
conclusion, but the bringing in, one by one, of the 



ARGUMENTS. 113 

instances bearing on the point in question, till a suf- 
ficient number has been collected 

" "We do not, strictly, reason by Induction, but reason 
from Induction : i. e., from our observations on one, or 
on several individuals, we draw a conclusion respecting 
a class they come under ; or, in like manner, from sev- 
eral Species to the Genus which comprehends them. 

2. The Inductive Syllogism an Enthymeme with a Sup- 
pressed Major. 

"We shall find that the expressed premise of the 
Enthymeme, viz. : that which contains the statement 
respecting the individuals is the minor; and that it is 
the major that is suppressed, as being in all cases sub- 
stantially the same; viz.: that what belongs to the 
, individual or individuals we have examined, belongs 
(certainly or probably, as the case may be) to the 
whole class under which they come. 

3. Necessity of Assuming a Major Premise. 

"It has, however, been urged that what are de- 
scribed as the major premises in drawing inferences 
from Induction, are resolvable ultimately into an as- 
sertion of the 'Uniformity of the laws of Nature,' or 
some equivalent proposition, and that this is, itself, 
obtained by Induction ; whence it is concluded that 
there must be at least one Induction — and that one, 
the one on which all others depend — incapable of be- 
ing exhibited in a syllogistic form. 

" But it is evident, and is universally admitted, that 
in every case where an inference is drawn from Induc- 
tion (unless that name is to be given to a mere ran- 
dom guess without any ground at all) we must form 

a judgment that the instance or instances adduced are 
I. in 



114 LOGIC. 

< sufficient to authorize the conclusion — that it ia 

< allowable' to take these instances as a sample war- 
ranting an inference respecting the whole class. 

"Now, the expression of this judgment in words, is 
the very major premise alluded to. To acknowledge 
this, therefore, is to acknowledge that all reasoning 
from induction, without exception, does admit of being 
exhibited in a syllogistic form ; and consequently that 
to speak of one Induction that does not admit of it is 
a contradiction. 

4. Origin of this Major Premise. 

" "Whether the belief in the constancy of Nature's 
l aws — a belief of which no one can divest himself — be 
intuitive and a part of the constitution of the human 
mind, as some eminent metaphysicians hold, or ac- 
quired, and in what way acquired, is a question 
foreign to our purpose." 

5. Objections. 

1st. The signification of induction given by Whate- 
ly — "the bringing in, one by one, of the instances' 
bearing on the point in question, till a sufficient num- 
ber has been collected"— is not the meaning generally 
attached to the word induction; for by induction is 
generally understood the inference that what is true 
of the parts is true of the whole. 

2d. Whately's suppressed major premise, " that 
what belongs to the individual or individuals we have 
examined, belongs (certainly or probably, as the case 
may be) to the whole class under which they come," 
is itself an induction, wider, as to its subject, than the 
induction expressed by the conclusion, and containing 
it as a particular case. 



ARGUMENTS. 115 

How does Whately account for this induction ? 

He says, "we must form a judgment that the in- 
stance or instances adduced are sufficient to authorize 

the conclusion Now the expression of this 

judgment in words is the very major premise alluded 
to." But how is this judgment, which is resolvable 
into a belief in the constancy of Nature's laws, ac- 
counted for? Whately replies, this "is a question 
foreign to our purpose." 

8. Mill's Theory of Induction. 

"Whatever be the most proper mode of expressing 
it, the proposition, that the course of nature is uni- 
form, is the fundamental principle or general axiom 
of Induction. It would yet be a great error to offer 
this large generalization as any explanation of the 
inductive process. On the contrary, I hold it to be 
itself an instance of induction, and induction by no 
means of the most obvious kind. Far from being the 
first induction we make, it is one of the last, or at all 
events, one of those which are latest in attaining phil- 
osophical accuracy Yet this principle, though 

so far from being our earliest induction, must be con- 
sidered as our warrant for all others, in this sense, that 
unless it were true, all other inductions would be fal- 
lacious." 

Remarks. 

1. As an explanation of formal induction in which 
the conclusion is demonstrably certain, this theory 
certainly fails ; for, Mr. Mill holds, 

1st. That the course of nature is uniform, is the 
fundamental principle or general axiom of Induc- 
tion. 



116 LOGIC. 

2d. That this principle is itself an induction of by 
no means the most obvious kind. 

Sd. That far from being the first induction we make, 
it is one of the last. 

4th. That this principle must be considered as our 
warrant for all others. 

Then it follows that all other inductions are with- 
out warrant, and that this principle is itself without 
warrant. 

2. As an explanation of real induction in which the 
conclusion is only probable, this theory is not liable to 
objection. 

3. The genesis of the induction, that the course of 
nature is uniform, is as follows: In a particular in- 
duction, the uniformity is observed in the cases exam- 
ined, and is, so far, positive. The uniformity observed 
in the cases examined affords a presumption of the 
same uniformity in the cases not examined, and this 
presumption is confirmed as experience enlarges. 

Many other inductions are formed and confirmed in 
the same way. 

At length we infer the grand induction that, The 
course of nature is uniform. 

4. Real induction affords no absolute certainty. 

" Even from the requisites of Induction and Anal- 
ogy, it is manifest that they bear the stamp of uncer- 
tainty; inasmuch as they are unable to determine 
how many objects or how many characters must be 
observed, in order to draw the conclusion that the 
case is the same with all the other objects, or with all 
the other characters. It is possible only in one way 
to raise Induction and Analogy from mere probability 
to complete certainty ; vi r ^., to demonstrate that the 



ARGUMENTS. 117 

principles which lie at the root of these processes, and 
which we have already stated, are either necessary 
laws of thought, or necessary laws of nature. To 
demonstrate that they are necessary laws of thought 
is impossible ; for Logic not only does not allow infer- 
ence from many to all, but expressly rejects it. 

" Again, to demonstrate that they are necessary laws 
of nature is equally impossible. This has, indeed, 
been attempted, from the uniformity of nature, but in 
vain. For it is incompetent to evince the necessity of 
the inference of Induction and Analogy from the fact 
denominated the law of nature; seeing that this law it- 
self can only be discovered by the way of Induction 
and Analogy. In this attempted demonstration there 
is thus the most glaring petitio principii. The result 
which has been previously given remains, therefore, 
intact: Induction and Analogy guarantee no perfect 
certainty, but only a high degree of probability, while 
all probability rests at best upon Induction and Anal- 
ogy, and nothing else." 

These remarks apply to real induction, not to formal. 



. 



9. True's Theory of Induction. 



Its office is to analyze phenomena, to mark the 
different qualities of objects, and to ascertain their 
precise effects ; but when you have certainly deter- 
mined what qualities in any case produce what effects, 
one single instance of causation is sufficient for the 
widest generalization. Show me the property of the 
magnet which attracts iron, and I hesitate not to pre- 
dict that whenever and wherever that quality appears, 
in like circumstances, it will be followed by the same 
effect Like causes produce like effects." 



118 LOGIC. 

Remarks. 

This explanation may be regarded correct as far as 
induction relates to the effects of like causes in like 
circumstances ; for whatever there is in the nature of 
the cause to determine an effect in certain circum- 
stances, a like cause in like circumstances, though 
numerically different, is, virtually, the same cause in 
the same circumstances, and the same result would 
follow. 

But induction is not restricted to the inference of 
effects from causes. 

Thus, a naturalist finds that sheep, cattle, deer, 
and all quadrupeds deficient in upper cutting teeth, so 
far as he has examined, ruminate; and thus concludes 
that all quadrupeds thus deficient ruminate. This 
conclusion may be universally true ; but can never be 
absolutely certain as long as there are quadrupeds in 
any part of the world that have not been examined. 

A man who has noticed that every human being 
observed by him had but one head, may infer that 
every human being has but one head ; yet this induc- 
tion might be overthrown by the next show exhibit- 
ing a human monstrosity with two heads. 

VII. DOCTRINE OF METHOD. 
1. Definition. 

1. Definition defined. 

A definition is such a description of an object as 
will distinguish it from all other objects. 

An object is defined by classing it under the genus 
immediately superior and giving the differential attri- 
bute which distinguishes it from its coordinates. 



DOCTRINE OF METHOD. 119 

2. Illustration. 

Thus, triangles, quadrilaterals, pentagons, etc., are 
coordinate species of the genus, polygon. Then we 
have the following definitions : 

A triangle is a polygon of three sides. 

A quadrilateral is a polygon of four sides. 

A pentagon is a polygon of five sides. 

3. Object. 

The object of definition is to distinguish the thing 
defined from other things, and thus to give clearness 
and precision to thought. 

4. Classification. 

Definitions are of three kinds, nominal, real, and 
genetic. 

1st. A nominal definition is a definition of a term. 
Thus, the word sphere signifies a volume bounded by 
a uniformly curved surface. 

2d. A real definition is a definition of a thing. 
Thus, a sphere is a volume bounded by a curved sur- 
face all the points of which are equally distant from a 
point within. 

3d. A genetic definition is a definition exhibiting 
the mode of producing the thing. Thus, a sphere is 
a volume generated by revolving a circle about a 
diameter. 

4. Laws. 

1st. The subject and predicate of a definition must 
be coextensive. Hence, the simple converse of a defi- 
nition is true. Thus, a triangle is a polygon of three 
sides, and conversely, a polygon of three sides is a 
triangle. 



120 LOGIC. 

2d. There are no exceptions to definitions ; for an 
exception would invalidate the definition. 

3d. A definition should be precise; that is, omit 
nothing essential and contain nothing unessential. 

4:th. A definition should be clear ; otherwise, it fails 
in its purpose. 

bth. A definition should not involve the circle ; that 
is, the predicate should not contain the name of the 
thing defined, any derivative of that name, or any 
term whose definition involves the thing defined. 

6th. A definition should not be made by means of 
negative or divisive attributes. 

1th. A definition should not involve a problematic 
judgment. 

5. Examples of Imperfect Definitions. 

1st. " Horses are four-footed animals/' The predi- 
cate is the definition of quadrupeds, the genus of 
which horse is a species. We then have the genus 
given but not the differential quality. The definition, 
therefore, violates the first law, since the predicate is 
much more extensive than the subject, including 
besides horses, sheep, cattle, lions, tigers, etc. 

2d. " Parallel lines are those which never meet.' 
This definition omits an essential point — that they lie 
in the same plane, otherwise they might never meet 
and yet not be parallel. 

3d. " Parallelograms are quadrilaterals whose oppo 
site sides are parallel and equal." This definition, 
though true, contains more than is necessary; for the 
definition would be complete without the words "and, 1 
equal," which should, therefore, be omitted as unes- 
sential. The equality of the opposite sides follows as 
a consequence of their parallelism. 



DOCTRINE OF METHOD. 121 

4th. "Net- work is any thing decussated or reticu- 
lated, with interstices between the intersections." 
This definition wants clearness ; for the terms em- 
ployed need defining more than the thing defined. 

bth. "A law is lawful command." This is a circular 
definition. So, taken together, are, " Quantity is any 
thing which may be made the subject of mathematical 
investigation;" "Mathematics is the science of quan- 
tity." 

6th. "Industry is not honesty." This does not tell 
what industry is, but what it is not. 

7th. " Patriotism is a moral, social, or political vir- 
tue." This is too indeterminate. 

8th. " A miracle is an effect or event contrary to the 
established constitution or course of things, or a sen- 
sible suspension or controlment of, or deviation from, 
the known laws of nature, wrought either by the im- 
mediate act, or by the concurrence, or by the permis- 
sion of God, for the proof or evidence of some partic 
ular doctrine, or in attestation of the authority of 
some particular person." Not content with the dis- 
crimination of a miracle from all other phenomena, 
the writer adds his theory of miracles, telling by 
whom wrought and for what purpose. 

2. Division. 

1. Definition. 

Division is the resolution of an extensive concept 
into the subordinate concepts contained under it. 

2. Object. 

1st. The primary object of division is to obtain ex 

tensive distinctness. 
l. 11 



122 LOGIC 

2d. The secondary object of division is to obtain 
completeness. 

3. Classification. 
1st. As to nature. 

a. Physical or real division, when the parts are act- 
ually separated. 

b. Metaphysical or ideal division, when the parts 
are separated in thought. 

2d. As to the number of coordinate members. 

a. A dichotomy, when there are only two coordinate 
members. 

b. A polytomy, when there are more than two co- 
ordinate members. 

a. A trichotomy is a polytomy of three members. 
p. A tetrachotomy is a polytomy of four members. 

4. The Principle of Division. 

The principle of division is that attribute in refer- 
ence to which the division is made. 

It is evident that the same class may be divided in 
reference to different principles. 

The following is an illustration : 

Triangles may be divided, 

(a. Scalene. 
1st. As to the sides, j & ^^ Bi-equilateral. 

\ P. Tri-equilateral. 

(a. Right angled. 
2d. As to the angles. J b. Oblique ( «. Acute angled. 
I angled. \/5. Obtuse angled. 



DOCTRINE OF METHOD. 123 



5. Laic. 

The Law of Contradictories regulates division. 

Thus, triangles, as to their angles, are right angled 
or non-right angled, that is, oblique angled. 

Strictly, every logical division gives a dichotomy. 
Sometimes, however, one of the two coordinate mem- 
bers is omitted and its parts substituted in its place. 

Thus, in dividing triangles as to their angles, in- 
stead of taking the two coordinate members, right 
angled and oblique angled, we may omit the oblique 
angled, and substitute in its place its parts — acute 
angled and obtuse angled. Thus, giving the follow- 
ing classification : 

( Right angled. 
Triangles. < Acute angled. 
^ Obtuse angled. 

6. Rules. 

1st. Every division should be made in reference to 
some principle. 

2d. Every division should be made in reference to 
only one principle. 

3d. The principle of division should be an actual 
determinate attribute of the divided class. 

4th. The principle of division should be selected 
with reference to the object to be accomplished. 

5th. Each of the members must be less than the 
class divided. 

6th. The sum of the members must be equal to the 
class divided. 

17th. The members must be mutually exclusive. 
8th. The members must be coordinate. 



124 LOGIC. 

9th. The divisions and subdivisions must proceed 
continuously; that is, each member must be immedi- 
ately subordinate to the concept under which it is 
placed. 

7. Faulty Divisions. 

1st. Human actions are necessary or free or useful 
or detrimental. 

2d. Triangles are divided into right angled, scalene, 
and acute angled. 

3d. Parallelograms are divided into squares and 
rectangles. 

4th. Human conduct is good or bad. 

bth. Philosophy is divided into theoretical, practi- 
cal, and moral. 

3. Analysis. 

1. Definition. 

Analysis is the resolution of a comprehensive con- 
cept into the attributes contained in it. 

2. Object. 
1st. The primary object of analysis is to obtain 
comprehensive distinctness. 

2d. The secondary object is to obtain completeness. 

3. Rule. 

Take the attributes common to all of the coordinate 
members of the genus immediately containing the 
species, or the species containing the individual, to- 
gether with the characteristic attributes of the species 
or individual to be analyzed. 

It will greatly aid in attaining comprehensive dis- 
tinctness to begin with being, the highest genus, and 



DOCTRINE OF METHOD. 120 

proceed by division, retaining the member containing 
the given concept under it, taking, at each step, the 
attributes of the genus, and the characteristics of the 
retained member, till the given concept is reached. 

4. Argumentation. 

1. Definition. 

Argumentation is a process of reasoning the object 
of which is to establish the truth of a proposition. 

2. Results. 

By argumentation truth is proved, the concatena- 
tion and dependence of thoughts are ascertained, and 
the congruence or harmony of thoughts is secured. 

3. Elements. 

1st. The conclusion or proposition to be proved. 
2d. The premises or grounds of proof. 
3d. The relation between the premises and the con* 
elusion. 

4. Conditions. 

1st. The premises must be known to be true. 
2d. The premises must be so related to each other 
and to the conclusion as to necessitate the conclusion. 

5. Classification. 
1st. As to validity. 

a. Valid, when the conclusion is proved. 

b. Invalid, when the conclusion is not proved. 

2d. As to the medium of proof. 

a. Mediate, when there is a middle term. 

b. Immediate, when there is no middle term. 



126 LOGIC. 

3d. As to form. 

a. Regular, when stated in due form. 

b. Irregular, when not stated in due form. 

4th. As to method. 

a. Direct, when the conclusion is proved directly. 

b. Indirect, when the conclusion is proved indi- 
rectly. 

5th. As to the order of procedure. 

a. Inductive, when the procedure is from the parts 
to the whole. 

b. Deductive, when the procedure is from the whole 
to the parts. 

6th. As to logical quantity. 

a. Extensive, when the reasoning is in extensive 
quantity. 

b. Comprehensive, when the reasoning is in com- 
prehensive quantity. 

7th. As to figure. 

a. Figured, when the terms are related as subject 
and predicate. 

b. Unfigured, when the terms are not related as 
subject and predicate. 

8th. As to the order of the premises and con- 
clusion. 

a. Analytic, when the conclusion is stated first. 

b. Synthetic, when the premises are stated first. 

9th. As to simplicity. 

a. Simple, when there is but one syllogism. 

b. Compound, when two or more syllogisms are 
combined. 



. 



DOCTRINE OF METHOD. 12? 

10th. As to cogency. 

a. Demonstrative, when the truth of the conclusion 
is absolute. 

b. Probable, when the truth of the conclusion is not 
absolute. 

11th. As to completeness. 

a. Complete, when all the parts are fully stated. 

b. Incomplete, when all the parts are not fully 
stated. 

12th. As to the nature of the premises. 

a. A priori, when the premises are intuitive prin- 
ciples. 

b. A posteriori, when the premises are established 
by experience. 

13th. As to the fundamental laws of thought in- 
volved. 

a. Categorical, when the form is determined by the 
Law of Identity or the Law of Conflictives. 

b. Hypothetical, when the form is determined by 
the Law of Reason and Consequent. 

c. Disjunctive, when the form is determined by the 
Law of Contradictories. 

d. Dilemmatic, when the hypothetic and disjunctive 
forms are combined. 

6. Rules. 

1st. Nothing is to be assumed to be true which is 
not known to be true; that is, "nothing is to be 
begged, borrowed, or stolen." 

2d. No proposition is to be employed as a premise 
the truth of which depends on the conclusion. 

3d. A proposition must not be used to prove Hself. 



128 LOGIC. 

4th. No leap or hiatus must be made. 
5th. A different proposition must not be proved in 
place of the given proposition. 

VIII. MODIFIED LOGIC. 

1. Truth. 

1. Definition. 
Truth is the harmony of thought with its object. 

2. Classification. 

1st. Formal truth, the harmony of thought with 
the form of thought. 

a. Logical truth, the harmony of thought with the 
necessary laws of thought. 

b. Mathematical truth, the harmony of thought 
with the necessary relations of quantity. 

2d. Real truth, the harmony of thought with its 
matter. 

a. Physical truth, the harmony of thought with ex- 
ternal phenomena. 

b. Metaphysical truth, the harmony of thought with 
the necessary facts of mind. 

c. Psychological truth, the harmony of thought 
with the contingent facts of mind. 

3. Criterion. 

The criterion of truth is intuition or demonstra- 
tion necessitating certainty; for, if certainty exists, 
all doubt is dispelled; for to doubt what we necessa- 
rily think is contradictory and impossible. 



MODIFIED LOGIC. 129 

2. Error. 

1. Definition. 

Error, is the opposite of truth ; and is, therefore, the 
want of harmony between thought and its object. 

2. Distinguished from Ignorance and Illusion. 

Ignorance is negation of knowledge, error is posi 
tive pretense to knowledge. 

An illusion is a deceptive appearance arising from 
certain conditions and affections in the thinking sub- 
ject. Thus, pressure on the eye causes spots to 
appear. 

3. Sources. 

1st. Ignorance, leading to the assumption of the 
non-existence of that of which we are ignorant. 

2d. Illusion, leading to the assumption that the de 
ceptive appearance is an objective reality. 

3d. The disturbing influence of the will or the sen- 
sibilities. 

4th. A defect in the object of knowledge. 

bth. Circumstances, nationality, social relations, ed- 
ucational prejudices. 

Qth. The constitutional peculiarities of the indi- 
vidual. 

7th. The defects inherent in language. 

8th. The nature of the knowledge about which 
thought is conversant. 

4. Remedies. 

1st. General intelligence. 

2d. A symmetrical and thorough education. 

3d, A proper application of logical principles. 



130 LOGIC. 

3. Investigation. 

1. Definition. 

Investigation is that intellectual process which has 
for its object the discovery of truth. 

2. Methods. 

Experience, observation, experiment, hypothesis, 
induction, analogy. 

1. Experience. 

1. Definition. 

Experience is the apprehension of external or inter- 
nal phenomena through perception and conscious- 
ness. 

2. Kinds. 

1st. Personal ; that is, our own experience. 
2d. Foreign ; that is, the experience of others. 
Personal experience is more certain ; foreign, more 
extensive. 

3. Relation to Knowledge. 
All knowledge begins with experience. 

2. Observation and Experiment. 

1. Definition. 

1st. Observation is the voluntary attention of the 
intellect directed to a certain object. 

2d. Experiment is an extension of observation ef- 
fected by means of instruments or apparatus by 
which we vary the circumstances of the phenomena. 



MODIFIED LOGIC. 131 

2. Conditions to be observed. 

1st. Subjective: The mind should be in vigorous 
condition, self-possessed, and free from prepossession, 
partiality, or prejudice. 

2d. Objective : The attention must be directed to 
the thing observed, which is to be divided and sub- 
divided, if necessary, till the perceptions become 
clear, and all foreign or adventitious matter is to be 
excluded. 

3. Rules for Procedure. 

1st. Observe, analyze, compare, and classify the 
phenomena. 

2d. Determine the conditions requisite to their 
reality, 

Sd. Ascertain the causes of the phenomena. 

4:th. Discover the laws of the phenomena. 

bth. Make a record of the results. 

4. Remark. 

A compliance with these rules will require also the 
application of the following methods of investigation. 

3. Hypothesis. 

1. Definition. 

An hypothesis is a supposition made to account for 
certain phenomena. 

2. Relation to the Phenomena. 

1st. An hypothesis that accounts for the phenom- 
ena may be regarded, provisionally, as the true ex- 
planation. 






132 LOGIC. 

2d. The only possible hypothesis that accounts for 
the phenomena must be regarded as the true ex- 
planation. 

L Real Induction. 

1. Definition. 

Heal induction is the process of reasoning from 
some of the parts to the whole. 

2. Varieties. 

1st. Individual induction; when the parts are indi- 
viduals of which the whole is the species. 

2d. Special induction ; when the parts are species 
of which the whole is the genus. 

3. Conditions. 

1st. The partial judgments from which the general 
judgment is inferred must be of the same quality; 
that is, all affirmative or all negative. 

2d. That a requisite number of parts be observed. 
This condition is necessarily somewhat vague, since 
the number required will vary with the circum- 
stances. 

4. Nature of the General Proposition. 

The general proposition inferred by induction is to 
be regarded only as probable. This probability is in- 
creased in proportion, 

1st. To the number and variety of the objects ob- 
served. 

2d. To the accuracy of the observations and com- 
parisons. 

M. To the clearness and precision of the agreement. 






MODIFIED LOGIC. 133 



4th. To the thoroughness of search for exceptions, 
none being found. 

15. Analogy. 
1. Definition. 
Analogy is the process of reasoning from the simi- 
larity of objects in certain respects to their similarity 
in other respects. 

2. Conditions. 

1st. The objects observed must agree in certain 
respects. 

2d. The attributes observed must not be all nega- 
tive or all accidental. 

3. Nature of the Conclusion. 

The conclusion is only probable. The probability 
of the conclusion is increased in proportion, 

1st. To the number and accuracy of the obser- 
vations. 

2d. To the number of congruent attributes. 

Sd. To the importance of the congruent attributes. 

4. Example. 

P has the attributes a, 6, c, and d. 

Q has the attributes a, 6, c. 

.-. Q probably has the attribute d. 

5. Refutation. 

This argument can be refuted if it can be shown 
either, 

1st. That d is the effect of some attribute which is 
in P but not in Q. 



134 LOGIC. 

2d. That there are present with P and absent from 
Q certain circumstances which are indispensable con- 
ditions of d. 

Sd. That Q has some attribute incompatible with d. 

4th. That the circumstances attending Q prevent the 
existence of d. 

6. Illustration. 

The earth is an opaque solid, nearly spherical, de- 
rives light and heat from the sun, and is inhabited. 

The moon is an opaque solid, nearly spherical, de- 
rives light and heat from the sun. 

.-. The moon is probably inhabited. 

The points of difference, that the moon is smaller, 
more rugged, revolves on its axis but once in 
twenty-eight days, has no atmosphere and no water, 
present a counter probability that the moon is not 
inhabited. 

If the points of agreement are equally likely to be 
the conditions of life, the probability that the moon 
is inhabited would vary directly as the number of 
such points of agreement. 

But, since some of the circumstances wanting on 
the moon, such as air and water, are indispensable 
conditions of life on the earth, we must conclude, 
either that the moon is not inhabited at all, or that 
the conditions on which life depends on the moon are 
totally different from the conditions on which life de- 
pends on the earth. The conditions of life on the 
moon being, therefore, different from those on the 
earth, if they exist at all, the more points of resem- 
blance established between the moon and the earth, 
the indispensable conditions which exist on the earth 






MODIFIED LOGIC. 135 

being wanting, the less the probability of the sup- 
posed different conditions, and, consequently, the less 
the probability that the moon is inhabited. 

7. Uses of Analogical Argument. 
If not refuted, it may be usefully employed, 
1st. To show the reasonableness of the conclusion. 
2d. To remove prejudice. 
3d. To silence objections. 
Ath. To prepare the mind for direct argument. 






8. Induction and Analogy compared. 

1st. By induction we infer that an attribute belong- 
ing to many objects of a class, belongs to all the 
objects of that class. 

2d. By analogy we infer that objects agreeing in 
certain respects, agree in other respects. 

3d. They agree in the fact that they give only prob- 
able conclusions, and that the degree of probability 
may vary between the limits, impossibility and cer- 
tainty, without ever reaching either limit. 

6. Examples of Investigation. 

1. Method of Agreement. 

The effect of A B C is a b c. 
The effect ofADEis^e. 

( b and c are not effects of A. 
d and e are not effects of A. 
a is not the effect of either B C or D E. 
a is the effect of A. 



136 LOGIC. 

2. Method of Difference. 

(The effect of A B C is a b c. 
Is*. < The effect of B C is b e. 
l.\ The effect of A is a. 

( The cause of a b c is A B C. 
2d. < The cause of 6 c is B C. 
^.\ The cause of a is A. 

3. Method of Residues. 

r A B C is the cause of a b c. 
B is the cause of 6. 
C is the cause of c. 
/. A is the cause of a. 

4. Method of Concomitant Variations. 

Let A', A" be variations of A, and a', a" variations 
of a. 

r A B C is the cause of a b c. 
A' B C is the cause of a f b e. 
A" B C is the cause of a n b c. 
\ A is the cause of a. 

This method is especially valuable in case of per- 
manent causes; that is, when cases can not be found 
free from their influence, as in the case of the connec- 
tion of the moon with the phenomena of the tides. 

Thus, though we can not remove the earth from 
the influence of the moon, yet we can observe the 
variations in the position of the moon with respect to 
the earth and the concomitant variations of the tides. 
The influence of the moon in the production of the 
tides is hence determined. 

In a similar manner the influence of the sun is 
determined. 



FALLACIES. 137 

IX. FALLACIES. 

Definition and Classification. 

A fallacy is an invalid intellectual process. 
There are three classes of fallacies, assumptions, 
sophisms, and aberrancies. 

1. Assumptions. 

1. Definition. 

An assumption is that which is taken as true with- 
out evidence. 

2. Ground of Invalidity. 

Assumptions may be true or false ; but, resting on 
no basis of evidence, they are, in both cases, invalid, 
not because known to be false, but because not known 
to be true. 

To assume an assumption false, because of its lack 
of evidence, would be a procedure as invalid as to 
assume it true. 

3. Classes. 

1st. Assumptions arising from non-observation or mal- 
observation. 

The common source of these is want of attention. 

Failing to notice many things, we are liable to as- 
sume their non-existence. 

Other things, not wholly overlooked, are, from in- 
attention, misapprehended, and, assuming them to be 
what they are not, we are involved in confusion. Of 
how many may it be said, " Having eyes, they see 
not." A very amusing and instructive example of 
this is found in a dialogue entitled "Eyes and No 
L. 12 



138 LOGIC. 

Eyes." Two persons passed over the same route the 
3ame day; but to one the journey was wholly devoid 
of interest, while to the other, objects of interest 
abounded on every hand, a glowing account of which 
he gave, greatly to the edification and astonishment 
of his unobserving friend. 

2d. Assumptions arising from prejudice. This is a 
fruitful source of assumptions, and it is very difficult 
to divest ourselves of its influence, yet we are loth 
to admit that we are, in any degree, subject to its 
control. " Can any good thing come out of Naza- 
reth?" is the expression of intensified prejudice. 

From too high an opinion of ourselves, too low an 
opinion of others, ruling desires, nationality, party, 
church, or society relations, education, and associ- 
ation, arise prejudices leading to assumptions which 
vitiate our judgments and involve us in error. 

The antidotes which neutralize the poison of preju- 
dice are, an honest heart, a good disposition, a love 
for truth, due caution, and patient investigation. 

3c/. Assumption that what is true of ourselves is true 
of others. How slow is a man habitually governed by 
selfish considerations to believe that another who has 
done a noble deed to bless humanity was actuated by 
disinterested motives ! How clearty is the true char- 
acter of a person often revealed by the judgment 
which he passes upon another! Conscious of his 
own dishonesty or impurity, he assumes that others 
are as dishonest and vile as he. 

4:th. Assumptions arising from superstition. Supersti- 
tion has produced a numerous brood from its mythol- 
ogies, oracles, omens, witchcrafts, apparitions, ghosts, 
fairies, signs, and charms. 

Though superstition is less potent now than for- 



FALLACIES. 139 

merly, it has not yet altogether lost its influence, as 
is indicated by such sayings as these : 

" If it rains the first Sunday of a month, it will rain 
every Sunday." 

"If you first see the new moon over your right 
shoulder, you will have good luck for that month." 

hth. Assumptions arising from hasty generalization. 
Finding of a certain nation a few individuals of a 
certain character, some hastily assume that to be the 
character of the whole nation. I well remember the 
time when I judged the whole English nation to be 
bigoted, narrow-minded, conceited, and overbearing, 
because that was the character of the few with whom 
I happened to be acquainted ; but I have since learned, 
that among the English are the large-hearted and 
noble, an honor to humanity and to God. 

The same fallacy is quite prevalent in reference to 
the subject of education. The failures of graduates, 
on the one hand, are pointed out as exhibiting the 
inutility of education, not recollecting that much 
worthless timber is sent to college, and yet, though 
not one of a thousand of the nation is a graduate of 
a college, the majority of the prominent public men 
have pursued the prescribed course of study, and have 
received the honor of graduation. 

On the other hand, the splendid achievements of a 
Franklin, or the sublime deeds of a Washington, are 
referred to as examples showing that education is not 
essential to the highest success, forgetting that these 
men were gifted with the noblest endowments, and 
that their struggles with difficulties developed their 
powers and enabled them to achieve immortal renown. 

But how rare such cases ! Shall we take such men, 
raised up and endowed by Providence to accomplish 



140 LOGIC. 

a great purpose, as indicating the principles which 
should guide us in the great work of education ? 
Then, although there might be a few great men, the 
mass of mankind would sink into the darkness of 
barbarism. 

This tendency to form hasty general conclusions is 
exhibited in other instances. Thus, a man confess- 
edly below par accidentally becomes rich. Forth- 
with his neighbors exclaim, " Fortune favors fools." 
They cast about for similar instances and are sure to 
be successful in finding or imagining them. They 
hence conclude that success is bestowed by a caprice 
of fortune, forgetting the multitude of instances in 
which ignorance and imbecility are attended with 
their legitimate fruits — poverty and degradation, and 
those cases in which intelligent, well-directed effort is 
crowned with the most triumphant success. 

The ground of the fallacy seems to be this : These 
cases are contrary to what might naturally be ex- 
pected, and, hence, their occurrence makes a strong 
impression, not likely to be forgotten, while the re- 
verse cases, though far more numerous, are looked 
upon as matters of course, and, making but a slight 
impression, are overlooked, and the general conclu- 
sion is drawn from the far less numerous, though 
more striking exceptional cases. 

6th. Assumption that appearances correspond to reali- 
ties. The ancients could not believe that the earth was 
round, or that it moved, because apparently contrary 
to the evidence of their senses. They could, how- 
ever, readily believe that the sun, and the moon, and 
the stars revolved round the earth every twenty-four 
hours, because such is the appearance. 

When the moon is viewed through flying clouds, 



. 



• 






FALLACIES. 141 



t seems to be moving and the cloud? seem at 
rest. But is that apparent motion the real motion 
of the moon ? On a certain occasion, a company of 
boys were looking at the moon through such fleecy, 
flying clouds. All, save one, decided that the moon 
was running away, and were deeply indignant at 
what they called his stupidity, when he affirmed that 
the apparent motion of the moon was owing to the 
real motion of the clouds. " Have n't we eyes ? 
Can't we see?" The other boy, with the sagacity of 
a philosopher, conducted his companions to a tree 
and directed them to look at the moon through the 
branches. To their utter astonishment, the moon 
maintained its position, and the clouds sailed far 
away. That boy became the great philosopher Gas- 
sendi. 

7th. Assumptions originating in preconceived opinions. 
When Copernicus advanced his theory respecting the 
motion of the earth, his opponents met him with the 
objection that, if it did move, a stone let fall from 
the top of a tower would not strike the ground at the 
foot of the tower, but at a little distance from it, in a 
direction contrary to the motion of the earth, just as a 
ball, as they affirmed, let fall from the mast-head, while 
the ship is sailing, does not strike the deck at the foot 
of the mast, but nearer the stern of the vessel. 

The Copernicaus met this objection, not by deny- 
ing the spurious fact, and proving, as they should 
have done, by direct experiment, that a ball let fall 
from the mast-head does not strike the deck nearer 
the stern than the foot of the mast, but by saying 
that the ball was no part of the ship, and that the 
motion forward was not natural either to the ship or 
to the ball, while, on the other hand, the stone was 






142 LOGIC. 

a part of the tower, and, therefore, the motions which 
were natural to the earth were natural to the stone, 
and, therefore, it should strike the ground precisely 
at the foot of the tower. 

The opponents of Copernicus were wrong in assum- 
ing that if the earth revolved, the stone would strike 
the ground a little distance from the foot of the 
tower in a direction contrary to the motion of the 
earth, whereas the reverse is true ; for the stone fall- 
ing from the top of the tower has the same motion 
eastward, and since the top of the tower is farther 
from the axis of rotation than the foot, its motion 
eastward is greater; consequently, the stone would 
fall a little to the east of the foot of the tower. 

The following experiments, the account of which 
is taken from Loomis's Astronomy, fully confirm the 
theory: The mean of twelve trials from a tower 
256 ft. high, at Bologna, gave .74 in. deviation to the 
east, and .47 in. to the south. The mean of thirty- 
one experiments from a tower 250 /£. high, at Ham- 
burg, gave .35 in. deviation to the east, and .11 in. to 
the south. The mean of one hundred and six trials 
at Freyburg, in a mine whose depth was 520/f., gave 
1.12 in. deviation east and .17 in. south. 

Prof. Loomis remarks: "The deviation south is not 
accounted for by the theory." But this deviation can 
be accounted for thus : The top of the tower, at any 
instant, is moving in a vertical plane tangent to 
the circle of latitude drawn through the foot of the 
tower, and since this tangent plane intersects the 
surface of the earth in a line which continually devi- 
ates to the south of the circle of latitude for all places 
north of the equator, and to the north for all places 
south of the equator, it follows that in north lati- 



FALLACIES. 143 

tude the deviation should be both east and south, in 
south latitude both east and north, and at the equator 
east only. 

The importance of testing theory by experiment, 
where it is practicable, is strikingly exemplified in 
the long prevailing doctrine of the ancients respect- 
ing falling bodies. According to the Aristotelians, 
p Heavy bodies must fall quicker than light ones; for 
weight is the cause of their fall, and the weight of the 
greater body is the greater." How easy it would 
have been, by direct experiment, to settle the question 
effectually ! A large stone and a small one might have 
been dropped, simultaneously, over a precipice, and 
they would have been found to strike the ground at 
the same instant. 

8th. Assumptions pertaining to space. By a rational 
mind unbiased by theory, space is apprehended as a 
Reality, independent and absolute. It is not appre- 
hended as a mere conception of the mind, or law of 
thought, nor as substance, material or spiritual, nor 
as the attribute of substance, nor as the object of 
creation or destruction. Reason apprehends space as 
infinite extension, the room for the universe. 

The idea of space, then, is a rational intuition, of 
which the idea of body is the chronological anteced- 
ent ; that is, in the order of time, the idea of body is 
developed in the intelligence before that of space. 
But, immediately, on the perception of body, reason 
apprehends the reality of space as its necessary logical 
antecedent. 

It is intuitively certain that if there is body, there 
must be room in which the body exists. 

We attain to the idea of space through that of 
body; but space itself, when once apprehended by 



144 LOGIC. 

reason, is known as absolute, infinite extension. 
Were all bodies swept from existence, there would 
still be space. Were all spiritual existences annihi- 
lated, space would still stretch forth in all directions, 
without limit, on, forever on, an infinite abyss. 

Certain philosophers, among whom is Hamilton, 
hold that the infinite is not an object of knowledge. 
Their doctrine is, to use a Hamiltonian expression, 
" The infinite is unthinkable" They resolve the notion 
of infinite space into a mental impotency to conceive 
that space has bounds. 

Hamilton holds that the idea of space involves con- 
tradictions. In order not to misrepresent his views, 
we quote his words : 

" Extension, then, may be viewed as a whole or as 
a part ; and in each aspect, it affords us two incogita- 
ble contradictories. 1°. Taking it as a whole : Space, 
it is evident, must either be limited, that is, have an 
end, a circumference; or unlimited, that is, have no 
end, no circumference. These are contradictory sup- 
positions ; both, therefore, can not, but one must, be 
true. 

"Now let us try positively to comprehend, posi- 
tively to conceive, the possibility of either of these 
two mutually exclusive alternatives. Can we repre- 
sent or realize in thought, extension as absolutely 
limited ? in other words, can we mentally hedge 
round the whole of space, conceive it absolutely 
bounded, that is, so that beyond its boundary, there 
is no outlying, no surrounding space? This is im- 
possible. Whatever compass of space we may inclose 
by any limitation of thought, we shall find that we 
have no difficulty in transcending these limits. Nay, 
we shall find that we can not but transcend them ; for 






FALLACIES. 1 45 

we are unable to think any extent of space except as 
within a still ulterior space, of which let us think, till 
the powers of thinking fail, we can never reach the 
circumference. We may, therefore, lay down the first 
extreme as inconceivable, w^e can not think space as 
limited. 

"Let us now consider its contradictory: can we 
comprehend the possibility of infinite or unlimited 
space? To suppose this is a direct contradiction in 
terms ; it is to comprehend the incomprehensible. 
We think, we conceive, we comprehend a thing, only 
as we think it within or under something else ; but to 
do this of the infinite, is to think the infinite as finite, 
which is contradictory and absurd." 

This view is here presented in its full force. To 
comprehend signifies to circumscribe. The infinite 
can not, therefore, be comprehended, for this would 
make it finite. But let us not be deceived by a word. 
Though the infinite is to be regarded as incompre- 
hensible, from the fact that it can not be referred to 
something greater under which it is classed, yet it 
does not follow that it is unknowable. 

The distinction between the comprehensible and 
the knowable is clearly shown by Hamilton himself. 
We quote from his Philosophy of Common Sense : 

" To make the comprehensibility of a datum of 
consciousness the criterion of its truth, would be, in- 
deed, the climax of absurdity. For, the primary data 
of consciousness, as themselves the conditions under 
which all else is comprehended, are, necessarily, them- 
selves incomprehensible. We know and can know, 
only that they are, not how they can be. To ask how 
an immediate fact of consciousness is possible, is to 
suppose that we have another consciousness, before 

L. 13 



146 LOGIC. 

and above that human consciousness, concerning 
whose mode of operation we inquire. Could we do 
this, verily, we should be as gods." 

This is clear and to the point. That is compre- 
hensible which can be referred to something else con- 
taining it. But intuitions, being ultimate, can not 
be referred to ulterior principles which comprehend 
them. They are, therefore, incomprehensible. But 
because incomprehensible, they are not, therefore, un- 
knowable; for, in the language of Hamilton, "we 
know that they are, not how they can be." 

But how do we know that space is infinite? This 
knowledge is not to be resolved into a mental impo- 
tency to conceive that space has bounds, but rather 
into a potency to apprehend that it can have no 
bounds ; for, as Hamilton has well said, " Whatever 
compass of space we may inclose by any limitation of 
thought, we shall find that we have no difficulty in 
transcending these limits. Nay, we shall find that 
we can not but transcend them ; for we are unable to 
think any extent of space except as within a still ulte- 
rior space." 

Space, then, has no limits beyond which there is 
not ulterior space; and as we can not but transcend 
any limit, we know that space is not finite. But, by 
the law of contradictories, one of two contradictory 
propositions must be true, and but one can be true. 
If, then, one is known to be false, the other is known 
to be true. But we know it to be false that space is 
finite ; therefore we know it to be true that space is 
infinite. 

In reference to the objection that we can not com- 
prehend the infinite, because that would make it 
finite, we reply that we indeed grant that we can not 



FALLACIES. 147 

form, in the imagination, a picture of the infinite ; for 
a picture necessarily has outlines or boundaries, or, 
in other words, is finite. But because the infinite can 
not be represented in the imagination, it does not 
follow that it can not be known by the reason. Our 
knowledge of the infinite, then, is not a representation 
by the imagination, but an intuition by the reason. 
Hamilton's contradictions thus arose from referring 
the infinite to the wrong faculty. 

If the genesis of the notion that space is infinite is 
the fact that it can not be conceived to be finite, then 
there is an equally good reason for the notion that 
space is finite, from the fact that it can not be con- 
ceived to be infinite. If one is a sufficient warrant for 
the notion that it is infinite, the other is a sufficient 
warrant for the notion that it is finite. Then we have 
sufficient warrant for the contradictory notions, that 
space is infinite, and that space is finite. 

The nearly universal notion that space is infinite 
must lead us to conclude that the origin of this 
opinion is not the fact that it can not be conceived 
to be finite; for then the notion ought to be equally 
universal that space is finite, from the fact that it can 
not be conceived to be infinite. 

The mere negation of power to conceive a proposition 
to be true is not a warrant for the inference that its 
contradictory is true ; but this warrant is found in the 
positive knowledge, gained by intuition, demonstration, 
or experience, that the proposition can not be true. 

Not only is there the want of power to conceive 
space as finite, which is, of course, a mere negation, 
an impotence, but there is positive power to see that 
it can not be finite. Then by the law of contradic- 
tories, it must be infinite. 



148 LOGIC. 

2. Sophisms. 

1. Definition. 
A sophism is an invalid argument. 

2. Classes. 

1st. Formal fallacies. These are, the undistributed 
middle, illicit process, negative premises, particular 
premises, affirmative premises and a negative conclu- 
sion, one negative premise and an affirmative con- 
clusion, one particular premise and a universal con- 
clusion, and ambiguous middle, which have already 
been discussed. 

2d. Petitio principii, the begging of the question 
This fallacy consists in deducing the conclusion from 
premises one of which depends on the conclusion. 
According to Mr. Mill, every syllogism involves 
this fallacy. He says : " It must be granted that in 
every syllogism, considered as an argument to prove 
the conclusion, there is a petitio principii. When we 
say, 

All men are mortal, 

Socrates is a man, 

.-. Socrates is mortal; 

it is unanswerably urged by the adversaries of the 
syllogystic theory, that the proposition, Socrates is 
mortal, is presupposed in the more general assump- 
tion, All men are mortal : that we can not be assured 
of the mortality of all men, unless we were previously 
certain of the mortality of every individual man ; that 
if it be still doubtful whether Socrates, or any other 
individual you choose to name, be mortal or not, the 



FALLACIES. 149 

same degree of uncertainty must hang over the asser- 
tion, All men are mortal : that the general principle, 
instead of being given as evidence of the particular 
case, can not itself be taken for true without excep- 
tion, until every shadow of doubt which could affect 
any case comprised within it is dispelled by evidence 
aliunde; and then what remains for the syllogism to 
prove? that, in short, no reasoning from generals to 
particulars can, as such, prove any thing: since from 
a general principle you can not infer any particulars, 
but those which the principle itself assumes as fore- 
known. 5 ' 

In grappling with this famous objection to the syl- 
logistic theory, we shall demonstrate that so far from 
it being true, " That the general principle, instead of 
being given as evidence of the particular case, can not 
itself be taken for true without exception, until every 
shadow of doubt which could affect any case com- 
prised within it is dispelled by evidence aliunde" it is 
a fact that the general principle, though not of course 
true unless every particular case included under it is 
true, is itself often established, in its utmost general- 
ity, without any reference to the particular cases in- 
volved. Do we establish the general principle that, 
Any term of an Arithmetical Progression is equal to the 
first term plus the number of the term minus one into the 
common difference, by examining all of the cases in- 
volved? Do we not, in fact, establish the general 
principle without reference to the particular cases ? 

Thus, let a denote the first term and d the common 
difference of any increasing arithmetical progression, 
then from the law of the series, the terms will be, 

a> a + J, a + 2 d, a + 3 d, a + 4 d, . . . 



i 



150 LOGIC. 

The coefficient of d in the second term being one, is 
one less than the number of the term, and since the 
coefficient of d increases by unity in the successive 
terms, it follows that this coefficient will always be 
less by unity than the number of the term; hence, 

The n th term = a + (n — l)d. 

Take the series, 3, 5, 7, 9, . . . 

Then, the 100** term = 3 + 99 x 2 = 201. 

The same formula will apply to an infinity of other 
cases, not one of which was taken into account in 
establishing the formula. Where is even the shadow 
of the petitio principii in this? The syllogism is liable 
to this charge only when the major premise is obtained 
by real induction, and then only apparently. 

Sd. Reasoning in a circle. This fallacy consists in 
assuming a premise involving the conclusion, and 
then from the conclusion deducing the premise. It 
is analogous to the petitio principii. An argument in 
which the premises involve the conclusion is not to 
be taken as a case of petitio principii, or reasoning in a 
circle, provided the premises be established by evi- 
dence independent of the conclusion. 

4:th. Saltus — leap in logic. This fallacy consists in 
suppressing one premise and inferring the conclusion 
from another with which it has no logical connection, 
on the assumption that the suppressed premise would 
justify the conclusion. 

5th. Fallacy of Division and Composition. The fal- 
lacy of division occurs when the middle term is used 
collectively in the major premise, distributive! y in the 
minor. 



FALLACIES. 151 

The Greeks overthrew Troy. 

Socrates was a Greek. 

•\ Socrates overthrew Troy. 

The fallacy of composition occurs when the middle 
term is used distributively in the major premise, and 
collectively in the minor. 

Three and four are two numbers. 
Seven is three and four. 
/. Seven is two numbers. 

6th. The fallacy that objects are incompatible because 
our conceptions of them are incompatible. M. Comte, 
the founder of the Positive Philosophy, says : " I 
must remark upon one very striking truth which 
becomes apparent during the pursuit of astronomical 
science — its distinct and ever-increasing opposition, 
as it attains a higher perfection, to the theological 
and metaphysical spirit. Theological philosophy sup- 
poses every thing to be governed by will, and that 
phenomena are, therefore, eminently variable, at least 
virtually. The positive philosophy, on the contrary, 
conceives them as subject to invariable laws, which 
permit us to predict with absolute precision. 

" The radical incompatibility of these two views is 
nowhere more marked than in regard to the phenom- 
ena of the heavens, since, in that direction, our pre- 
vision is proved to be perfect. The punctual arrival 
of comets and eclipses, with all their train of minute 
incidents, exactly foretold, long before, by the aid of 
ascertained laws, must lead the common mind to feel 
that such events must be free from the control of any 
will, which could not be will, if it was thus subordi- 
nated to our astronomical decisions." 



152 LOGIC. 

Let us see if the power of prevision is so fatal to 
theological conceptions as M. Comte would have us 
believe. 

He assumes that a will is necessarily variable and 
capricious, as is the case to a greater or less extent 
with respect to the will of man. The phenomena of 
the heavens are certainly subversive of the idea that 
the universe is governed by such a will. But it is a 
theological conception that with God " is no variable- 
ness, neither shadow of turning." How is the uni- 
formity of the astronomical laws incompatible with 
the idea of a God who is " the same yesterday, to-day, 
and forever?" 

It is a theological conception that God created the 
material universe as the theater on which should act 
his intelligent creatures. Now, the stability of the 
material universe, consequent upon the uniformity of 
the laws of nature, is essential to the continued exist- 
ence of the inhabitants of the world as they are at 
present constituted. Hence, since God wills the ex- 
istence of man on earth, he also wills the uniformity 
of the laws of nature, and this uniformity, which is 
an indispensable condition of the act of prevision, is 
not subversive of theological conceptions, nor incom- 
patible with the idea of a^ God. 

Comte makes one qualification which destroys the 
supposed incompatibility between astronomical and 
theological science. He says that theological philos- 
ophy, which represents every thing as governed by 
will, makes phenomena variable, "at least virtually;" 
but this is perfectly consistent with the fact that acta* 
ally j at least for an indefinite period, phenomena may 
be invariable. 

But this uniformity, though observed through a 



FALLACIES. 153 

long period, does not prove that things will always 
remain as they are at present constituted. 

The invariable uniformity of astronomical phenom- 
ena, through the period of human history, is perfectly 
compatible with the sublime declaration, " Of old hast 
thou laid the foundations of the earth, and the heav- 
ens are the work of thy hands. They shall perish, 
but thou shalt endure : yea, all of them shall wax old 
like a garment; as a vesture shalt thou change them, 
and they shall be changed. But thou art the same, 
and thy years shall have no end." 

The unchangeableness of God consists in his pur- 
poses which never vary, and is perfectly consistent 
with the fact that he carries out his plans, in a pro- 
gressive series of acts to their final consummation. 

Again, in reference to Physics, M. Comte says : 
" With this science begins the exhibition of human 
power in modifying phenomena. In astronomy, hu- 
man intervention was out of the question — in physics, 
it begins; and we shall see how it becomes more pow- 
erful as we descend the scale. 

"This power counterbalances that of exact previ- 
sion which we have in astronomy. The one power or 

the other — the power of foreseeing or of modifying 

is necessary to our outgrowth of theological philos- 
ophy. Our prevision disproves the notion that phe- 
nomena proceed from a supernatural will, which is 
the same thing as calling them variable; and our 
ability to modify them shows that the powers under 
which they proceed are subordinate to our own. 

"As the phenomena of any science become more 
complex, the first power [that of prevision] decreases, 
and the other [that of modifying] increases, so that 
the one or the other is always present to show 



154 LOGIC. 

unquestionably, that the events of the world are not 
ruled by supernatural will, but by natural laws." 

But how does man's power of modifying the cir- 
cumstances which surround him disprove the fact 
of a supernatural will ? It is God's will that man 
" should have dominion over the fish of the sea, and 
over the fowl of the air, and over the cattle, and over 
all the earth, and over every creeping thing that 
creepeth upon the earth." This certainly gives man 
ample license to modify the circumstances which sur- 
round him, and cause them to subserve his interests 
and promote his happiness, and this modification is 
not subversive of theological conceptions nor incom 
patible with the idea of a God. 

It will be observed that we have not assumed the 
superfluous task of proving the validity of theological 
conceptions or the truth of the being of a God, but 
have shown that Comte has failed to prove the in- 
compatibility of positive philosophy with theological 
science. 

3. Aberrancies. 

1. Definition. 

An aberrancy is a wandering from the conclusion 
warranted by the premises and drawing another un- 
warranted. 

2. Classes. 

1st. Inferring the conclusion false because the premises 
are false or the reasoning illogical. In such cases the 
proper inference is not that the conclusion is false, but 
that it is not proved. 



FALLACIES. 155 

Suppose a student should fail in his attempt to dem- 
onstrate the proposition that, The square of the hypoth- 
enuse of a right-angled triangle is equal to the sum of the 
squares of the other sides, would his failure invalidate 
the proposition? 

Aberrancies of this kind are not unfrequent, and 
are even committed by experienced debaters. Thus, 
Dr. Rice, in his debate with Mr. Pingree, says: "I 
have undertaken to prove the conclusion false, by 
showing the premises on which it is based to be un- 
sound. Is this not a fair mode of reasoning? If the 
premises are false, the conclusion can not be true. . 
If the principle be false, the conclusion based upon it 
is certainly false." Dr. Rice should have said, Mr. 
Pingree has failed to prove his proposition. The con- 
clusion may be objectively true, though one or both 
of the premises be false, as is seen in the following 
case : 

Every month has thirty days. 

April is a month. 

.-. April has thirty days. 

2d. Inferring the reasoning valid because the conclusion 
is true. 

It does not follow, because the conclusion is true, 
that the argument is valid. Many an unsound argu- 
ment has escaped detection, because the conclusion 
of the speaker coincided with the opinions of the 
hearers. 

3d. Ambiguity in the conclusion. This occurs when 
the conclusion is susceptible of two interpretations, 
one of which is a legitimate deduction from the prem- 
ises and the other not. The conclusion is regularly 



156 LOGIC. 

drawn and all seems fair; but the reasoner intention- 
ally or unconsciously passes to the second interpreta- 
tion, and claims that as the legitimate conclusion. 
Thus, in the following example : 

Whatever is foreknown must be as foreknown. 

Human volitions are foreknown. 

.*. Human volitions must be as foreknown. 

But this may mean, simply, that it must be true, as 
a matter of fact, that human volitions are as they are 
foreknown ; or that human volitions are necessitated 
to be just as they are — that they can not, by any pos- 
sibility, be otherwise. 

Whatever A knows must be as he knows. 
A knows that B is present. 
.-. B must be present. 

A thing must be as known, because it must be 
known as it is, so far as it is known at all. The as- 
sumption of the fact corresponding to the knowledge 
is a logical necessity to account for the knowledge ; 
but implies nothing in regard to necessity in the 
thing itself. The fact might have been different, then 
the knowledge would also have been different. 

4:th. The fallacy of objections. This consists in reject- 
ing, as false, that which is liable to objection. The 
atheistic argument drawn from the fact of moral evil 
can be thus stated : 

If God had been both willing and able to prevent 
sin, it would not have occurred; but sin has occurred ; 
.*. God is either able and not willing to prevent it, 
which is inconsistent with his holiness; or willing 
and not able, which is inconsistent with his omnipo- 
tence ; or neither willing nor able, which is inconsist- 



FALLACIES. 157 

ont both with his holiness and his omnipotence ; but 
either of these consequences is destructive of the idea 
of a God ; .-. there can be no God. 

In reply it may be said that God, if he had seen fit, 
might have created a universe in which all moral evil 
should be excluded forever. But from such a uni- 
verse, though displaying infinite perfection in its 
mechanism, all moral excellence would also be ex- 
cluded ; for, since necessitated action possesses no 
moral character, moral excellence implies liberty, and 
liberty involves the possibility of moral evil. Hence, 
to the mind of God, three alternatives were presented : 
No universe at all, or a mechanical universe in which 
all disorder and all moral excellence should be ex- 
cluded, or a moral universe in which both moral evil 
and moral excellence should be possible. Who can 
affirm that the latter alternative was not preferable ? 
Because the omniscient God chose to create a moral 
universe, shall short-sighted human reason deny his 
holiness or his omnipotence? 

4. Examples of Fallacies. 

Let the student point out the fallacies in the fol- 
lowing examples : 

"All good fathers provide food and clothing for 
their children. 

1. < Mr. B. provides food and clothing for his 

children. 
^.\ Mr. B. is a good father. 

f All moral beings are accountable. 

2. \ No brute is a moral being. 

^.\ No brute is accountable. 



158 LOGIC. 



3. 



9. 



r Those who found universities are patrons of 



learning. 



King Alfred founded the University of Oxford. 
..•. King Alfred was a great scholar. 

f No pagan is a Christian. 
4. \ Every villager is a pagan. 
^.\ No villager is a Christian. 

{]STo cat has nine tails. 
Every cat has one tail more than no cat. 
/. Every cat has ten tails. 



6. < 



8. 1 



r That side of the river is one side of the river. 

This side of the river is not that side of the river. 

.-. This side of the river is the other side of the 
v. river. 

If Christianity were from God, it would be uni- 
versal. 
It is not universal. 
l.\ It is not from God. 

^The fact of knowledge implies a correspond- 
ence of nature between the knowing subject 
and the known object. 
Matter is to mind an object of knowledge. 
.-. Mind is resolvable into matter, or matter 
into mind. 



r Things totally unlike can not act upon each 
other. 
Mind and matter are totally unlike. 
l.\ Mind can not act upon matter. 



FALLACIES. 159 

f Mind can not act upon matter. 
The movements of the body correspond to the 
10. < relations of the mind. 

•. There is a preestablished harmony between 
mind and matter. 



11. i 



r The volitions of the mind can not cause the 

movements of the body. 
The hypothesis of a preestablished harmony is 

untenable. 
.-. God causes the movements of the body to 
correspond with the volitions of the mind. 



12. Euathlus, a young man, agreed to pay Protag- 
oras, the prince of sophists, a large sum of money to 
accomplish him as a legal rhetorician. One half the 
sum was paid down, and it was agreed that the other 
half should be paid on the day when Euathlus should 
plead and gain his first case. But as the scholar was 
not in so much of a hurry to commence his legal 
practice as the master to obtain the other half of his 
fee, Protagoras brought Euathlus into court, and 
addressed him thus : " Learn, most foolish of young 
men, that, whether the judges decide in your favor or 
against you, pay me my demand you must. For if 
the judgment be against you, I shall obtain the fee 
by decree of the court, but if in your favor, I shall 
obtain it by the terms of the contract, since it be 
comes due on the very day you gain your first case." 
To this Euathlus rejoined: " Learn, most sapient of 
masters, from your own argument, that whatever 
may be the finding of the court, absolved I must be 
from any claim by you. For if the decision be favor- 
able, I pay nothing by the sentence of the judges, but 






lt)0 LOGIC. 



if unfavorable, I pay nothing in virtue of the compact, 
since, though pleading, I shall not have gained my 
case." 

13. "It is an observation which all the world can 
verify, that there is nothing so deplorable as the con- 
duct of some celebrated mathematicians in their own 
affairs, nor any thing so absurd as their opinions on 
the sciences not within their jurisdiction." 

Hence, the study of mathematics destroys common 
sense. 

14. There are in all of the professions, even distin- 
guished men, who hold the most absurd opinions on 
sciences not within their jurisdiction. 

Hence, the practice of any profession destroys com- 
mon sense. 




ANALYSIS OF CONTENTS. 



PA3ES. 

Preface 3 

Introduction 7 

Intuitions. . . . . . • . . • . ,7 

Definition, Classification. . . • • 7 

Conditions, Relations, Order of Evolution. . . . .8 

Propositions. .......... 9 

Corollaries. . .. . .10 

Thoughts 10 

Definition, Processes, Products. . 10 

Contents, Expression 11 

Science 11 

Definition, Classification 11 

Logic 13 

General Outline. ........ .13 

Definition, Exposition, Classification. 13 

Congruence, Confliction, Opposition. . . « , . . 15 

Definitions, Classification, Formulas 15 

Fundamental Laws of Thought. . . . . # . . 16 

The Laws of Identity . . 16 

The Law of Connectives. 16 

The Law of Contradictories. ........ 16 

i, 14 161 



162 LOGIC. 

PAGE. 

The Law of Reason and Consequent. . . 17 

Concepts ... 17 

Definition, Etymology, Nature of the Elements. ... 17 

Formation, Relation to language, Characteristics 18 

Classification as to Quantity, Extension, and Comprehension. . 19 

Classification as to Quality, Distinctness 21 

Concepts admitting Extensive or Comprehensive Distinctness. . 22 

Specific Rules for attaining Distinctness. 22 

Sources of Indistinctness, Remedy for Indistinctness. . . 22 

Classification as to Validity, Truth, Congruity, Completeness. . 23 

Classification as to Relations in Extension 24 

Notation expressing these Relations. .... 25 

Summary of the Relations of Extensive Concepts. • . . . 26 
The Laws of Classification by Genera and Species. . . .27 

Classification as to Relations in Comprehension. ... 28 

Judgments. 29 

Definition, Expression, Elements. 29 

Concepts and Judgments Compared 29 

Classification as to Origin, Validity, and Truth. ... 30 
Classification as to Extension and Comprehension. . . .30 

Classification as to Form, Quantity, and Quality. ... 31 

Principles of Expression ... 32 

Principles warranting Affirmation 32 

Principles warranting Negation. . . . . . 32 

Principle warranting Hypothecation 32 

Principle warranting Disjunction. ....... 32 

Classification of Categorical Judgments 33 

Laws of Validity ... 33 

Opposition 34 

The Laws of Opposition .35 



ANALYSIS OF CONTENTS. 163 

PAGE. 

Distribution of the Concepts of a Judgment 35 

Definitions, Principles, Consequences 35 

Conversion, Definition, Kinds 36 

Classification of Hypothetical Judgments 38 

Classification of Disjunctive Judgments 40 

Classification of Logical Disjunctives 41 

Classification of Dilemmatic Judgments. . . . • • '. 41 

Arguments, Definition. ......... 42 

Immediate Arguments, Definition, Varieties. .... 42 

Mediate Arguments, Definitions, Expression 44 

Illustration. ........... 45 

Remarks on Mediate Arguments. ....... 47 

Laws warranting the Conclusions v . . . 48 

Formal Fallacies 50 

Undistributed Middle. 50 

Illicit Process. 53 

Particular Premises 54 

Negative Premises . .55 

An Affirmative Conclusion and one Negative Premise. . . 56 
A Negative Conclusion from Affirmative Premises. . . .56 

A Universal Conclusion and a Particular Premise. . . . 57 

Ambiguous Middle .57 

Rules 57 

General Laws of the Syllogism 58 

Figure, Definition 58 

Classification 59 

Mood, Definition, Remark 59 

Positive Determination of the Valid Moods 60 

The Number of the Valid Moods 61 

Negative Determination of the Valid Moods 62 



164 LOGIC. 

PAGE. 

Figure I ... 64 

Valid Moods in Figure 1 64 

Doctrine of Figure 1 65 

Aristotle's Dictum .... 65 

Arguments in Figure I, with their Names 65 

Figure II 66 

Valid Moods in Figure II. 66 

Doctrine of Figure II 67 

Arguments in Figure II, with their Names 67 

Figure III 68 

Valid Moods in Figure III 68 

Doctrine of Figure III 69 

Arguments in Figure III, with their Names. .... 69 

Figure IV 70 

Valid Moods in Figure IV 70 

Doctrine of Figure IV 71 

Arguments in Figure IV, with their Names. . . . . 71 
Summary of the Names of the Arguments. . . . .72 

Signification of the Consonants in Figures II, III, IV. . . 72 

Direct Reduction 73 

Indirect Reduction 74 

Examples. 78 

Hypothetical Syllogisms, Definition, Examples. ... 80 

The propositions of an Hypothetical Syllogism 80 

Laws, Categorical and Hypothetical Syllogisms Compared. . 81 
Reduction of Hypothetical Syllogisms to Categorical . . .82 

Disjunctive Syllogisms, Definition, Examples. .... 82 
The propositions of a Disjunctive Syllogism. . . . .82 

Disjunctive Syllogisms of two Members 83 

Laws . . 83 



ANALYSIS OF CONTENTS 166 

PAGE. 

Disjunctive Syllogisms of more than two Members ... 83 
Laws. . . . . ... 84 

Categorical and Disjunctive Syllogisms Compared. . . 84 

Dilemmatic Syllogisms, Definition. 84 

Forms, Eemark 85 

Enthymemes, Definition, Etymology, Examples 86 

Prosyllogism and Episyllogism, Definitions 86 

Example. 87 

Sorites, Definition, Forms 87 

Laws 88 

Expansion of the Sorites • . . . 89 

The Epichirema, Definition, Etymology, Examples. . . .89 
Unfigured Syllogism, Definition, Examples, Laws. ... 90 

Reductio ad Absurdum, Definitions 90 

Principles, Application. 91 

Example 92 

The Exhaustive Method, Definition 93 

Compared with the Reductio ad Absurdum, Example. . . .93 
Hamilton's Notation and Classification of Propositions. . . 94 

Converse, Laws of Validity. . 95 

Opposition of Judgments 96 

Thompson's Criticism on (??) and (w). . . . . .98 

Hamilton's Reply. ... ... .99 

De Morgan's Criticism on (w). . • . • . . 100 

Hamilton's Reply 100 

Hamilton's Scheme of Figured Syllogisms 101 

Explanatory Remarks. 101 

Figure I. .102 

Figure II. ..103 

Figure III 104 



166 LOGIC. 

PAGE. 

Indeterminateness of Language. ... . . 105 

Positive Propositions 106 

Positive Syllogisms. 107 

Induction. ........... 109 

Definition, Classification, Position in Logic 109 

Logical Induction, Definition, Examples 109 

Law 110 

Mathematical Induction, Definition, Example 110 

Day's Theory of Induction Ill 

Objection to this Theory .112 

Whately's Theory of Induction 112 

Objections 114 

Mill's Theory of Induction, Remarks .115 

True's Theory of Induction. . . • 117 

Remarks. 118 

Doctrine of Method. ,118 

Definition, Definition defined 118 

Illustration, Object, Classification, Laws 119 

Examples of Imperfect Definitions 120 

Division, Definition, Object 121 

Classification, The Principle of Division 122 

Law, Rules 123 

Faulty Divisions 124 

Analysis, Definition, Object, Rule. .... 124 

Argumentation, Definition, Results, Elements 125 

Conditions, Classification 125 

Rules 127 

Modified Logic 128 

Truth, Definition, Classification, Criterion 128 

Error, Definition, Sources, Remedies. . . ... 129 



ANALYSIS OF CONTENTS. 167 

PAGE. 

investigation, Definition, Methods 130 

Experience, Definition, Kinds, Relation to Knowledge. . . 130 

Observation and Experiment, Definition 130 

Conditions, Rules, Remark 131 

Hypothesis, Definition, Relation to the Phenomena. . . . 131 
Real Induction, Definition, Varieties, Conditions. . . . 132 
Nature of the General Proposition. . . .. . . .132 

\nalogy, Definition, Conditions 133 

Nature of the Conclusion, Example, Refutation 133 

Illustration 134 

Uses of Analogical Argument. 135 

Induction and Analogy Compared 135 

Examples of Investigation 135 

Method of Agreement. 135 

Method of Difference 136 

Method of Residues 136 

Method of Concomitant Variations. 136 

Fallacies , . .137 

Definition and Classification 137 

Assumptions, Definition, . 137 

Ground of Invalidity. . . .... 137 

Classes 137 

Assumptions arising from non-observation or mal-observation. . 137 

Assumptions arising from prejudice. 138 

Assumption that others are like ourselves 138 

Assumptions arising from Superstition 138 

Assumptions arising from hasty Generalization. . . . 139 

Assumption that Appearances correspond to Realities. . . 140 
Assumptions originating in Preconceived Opinions. . . .141 
Assumptions pertaining to Space 143 



168 LOGIC. 

PAGE. 

Sophisms, Definition, Classes. 3 48 

Formal Fallacies. 148 

Petitio Principii 148 

Reasoning in a Circle. 150 

Saltus — leap in logic 150 

Fallacy of Division 150 

Fallacy of Composition. . 151 

The fallacy that objects are incompatible because our conceptions 

of them are incompatible. 151 

Aberrancies, Definition, Classes. 154 

Inferring the Reason invalid because the Conclusion is True. . 155 

Ambiguity in the Conclusion 155 

The Fallacy of Objections 156 

Examples of Fallacies. 167 










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